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Multi-Robot Task Allocation Based on

Swarm Intelligence

Shuhua Liu1, Tieli Sun1 and Chih-Cheng Hung2

1Northeast Normal University

2Southern Polytechnic State University

1China

2USA

1. Introduction

In the field of cooperative robotics, task allocation is an issue receiving much attention.

When researchers design, build, and use cooperative multi-robot system, they invariably try

to answer the question of which robot should execute which task. This is in fact a multi-

robot task allocation problem (MRTA). The task allocation problem addresses the question

of finding the task-to-robot assignments that optimize global cost or utility objectives.

Finding an optimal task allocation, even in a relatively simplified case, is an NP-hard

problem. Therefore, the majority of common approaches are approximate or heuristic in

nature. Those approaches usually give suboptimal solutions. MRTA is a fundamental issue

of the multi-robot systems, which embodies the high-level system organization and

operation mechanism. The quality of task allocation algorithm directly affects the

performance of multi-robot system. With an increase in the number of robots and difficulty

of tasks within a system, the issue of task allocation has risen to prominence and become a

key research topic in the multi-robot domain. In 2005, the International Conference on

Robotics and Automation (ICRA 2005) set special panels on multi-robot task allocation, in

which the latest research and the progress are discussed.

Gerkey and Mataric (2004) presented a particular taxonomy for the task allocation problem.

It is described as follows:

•

Single-task robots (ST) vs. multi-task robots (MT): ST means that each robot is capable

of executing at most one task at a time, while MT means that some robots can execute

multiple tasks simultaneously.

•

Single-robot tasks (SR) and multi-robot tasks (MR): SR means that each task requires

exactly one robot to achieve it, while MR means that some tasks can require multiple

robots.

•

Instantaneous (IA) and time-extended (TA) assignment: In the instantaneous

assignment, robots do not plan for future allocations and are only concerned with the

one task they are carrying out at the moment (or for which they are considering

executing). In the time-extended assignment, robots have more information and can

come up with longer-term plans involving task sequences or schedules.

Based on above categorization, there are eight types of task allocation combination. ST-SR-

IA is the simplest, as it is actually a trivial instance of the Optimal Assignment Problem

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394

(OAP). ST-MR-IA often appears in real world applications; that is, some tasks require the

combined effort of multiple robots. These two types of tasks are also called loosely-coupled

tasks and tightly-coupled tasks, respectively. Although some approaches for solving either

loosely-coupled task or tightly-coupled task allocation have been proposed, few approaches

for solving both loosely-coupled and tightly-coupled task allocation have been developed.

In this chapter, we present a task allocation mechanism based on swarm intelligence for the

large-scale multi-robot system, with both loosely-coupled and tightly-coupled task

allocation. The mechanism adopts a hierarchical architecture. At the high level, we employ

an Ant Colony Algorithm to find optimal allocations. Namely, each ant performs a task

allocation so as to choose an undertaker for every task. At the low level, each ant forms a

task-oriented robot coalition to perform a tightly-coupled task. Ant colony optimization

(ACO), the particle swarm and ant colony optimization (PSACO) and the quantum-inspired

ant colony optimization (QACO) are adopted to form the coalition. Finally, the algorithm is

implemented in the TeamBots simulation platform. Simulation results show that the

proposed mechanism can effectively solve loosely-coupled and tightly-coupled task

allocation in the large-scale multi-robot system.

2. Related work

Recently a number of solutions have been proposed in the literature to MRTA problems

(Zhang & Liu, 2008). These include behaviour based approaches such as ALLCANCE (Parker,

1998), BLE (Werger & Mataric, 2000) and ASyMTRe(Tang & Parker, 2005). The advantage of

these approaches possesses real-time, fault-tolerance and robustness; the solution, however,

can only be locally optimal. The market-based approach is the current mainstream of task

allocation methods. The representative method is CNP (Contract Network Protocol) which

proposed by Smith (1980). Other typical examples include First-price auctions (Zlot et al, 2002),

Dynamic Role Assignment (Chaimowicz et al, 2002), Traderbots (Dias, 2004), M+ (Botelho &

Alami, 1999), MURDOCH (Gerkey & Mataric, 2002a) and DEMiR-CF (Sanem & Tucker, 2006).

Because of better scalability, this method is particularly well-suited to the distributed robotic

domain. Furthermore, it is guaranteed to produce optimal allocations, but robots must

cooperate through explicit communication and more resource consumption. Once the

communication is interrupted, the performance of this method will degrade significantly

(Kalra & Martinoli, 2006). Therefore, it is suitable for small- to medium- scale task allocation

problems. Derived from the behaviours of social insects, the swarm intelligence approach is

exhibiting several good features such as self-organizing ability in unknown environments, and

emergent and adaptive behaviours through simple interaction among individuals. Since

cooperative individuals are distributed and there is no central control and global data in the

group, the system will be more robust. The failure of one or several individuals will not affect

the whole solution. Additionally, individuals cooperate through implicit communications. As

the number of the individuals in the system increases, the amount of communication grows

quite slowly. Therefore the swarm intelligence approach is the most suitable for distributed

multi-robot systems and as such more and more researchers have applied it to the multi-robot

task allocation, especially in dynamic environments. Ding et al. (2003) and Yang &Wang (2004)

adopted Ant colony algorithm for multi-robot cooperation. Zhang et al. (2007) employed

swarm intelligence for adaptive task assignment. Zhang & Liu (2008 b, 2009) and Liu & Zhang

(2009, 2010) conducted intensive research on swarm intelligence and applied it to the task

allocation of large-scale multi-robot system.

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3. Architecture

Ant Colony Algorithm is a new intelligent optimization algorithm and first proposed by

Colorni et al. (1992). In ant colony algorithm, each ant searches for solutions

independently in the candidate solution space, and lays some pheromone on the found

solution. The better the solution, the more pheromone the ant lays. A solution with higher

pheromone has a much greater chance of being chosen, and consequently this gives a kind

of positive feedback. Through this positive feedback, ants can eventually find the optimal

solution. Via this process the algorithm effectively solves combinatorial optimization

problems and performs especially well in solving complicated problems (Jiang et al, 2003;

Xia et al, 2005).

The paper adopts a hierarchical architecture, as shown in Fig.1. At the high level, we employ

the Ant Colony Algorithm to find optimal allocations. Let an ant denote a task; each ant

forms its task allocation so as to choose an undertaker for every task. At the low level, each

ant forms a task-oriented robot coalition to perform a tightly-coupled task by the ant colony

optimization (ACO), the particle swarm and ant colony optimization (PSACO) and the

quantum-inspired ant colony optimization (QACO). It is worth mentioning that the

proposed mechanism can not only solve loosely-coupled task allocation, but also tightly-

coupled task allocation because ants in the high level denote tasks instead of individual

robots. Finally, simulation results give a performance comparison, and then conclusions

follow.

ACO based task allocation

Low-level

coalition

formation

High-level

Task allocation

Task 1

R2

R

Rm

…

coalition

formation

R1

Rn

…

Task N

R1

Rm

…

coalition

formation

R2

Rn

…

Fig. 1. Hierarchical architecture of the system

4. Key issues of robot coalition formation

4.1 Validity of robot coalition

Similar to agent coalition formation, robot coalition formation also tries to find the robot

coalition with the greatest value that can complete a task t. A coalition may be formed by

several arbitrary robots in the system. However, in order to obtain a satisfactory result, we

must consider all or most of the combinations. Therefore it is a complex combinatorial

optimization problem. In addition, although there are many similarities between agent

coalition and robot coalition, there are also inherent differences which should not be

overlooked.

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Firstly, software agents are simply code fragments whose capabilities corresponding to

software functionality and current data knowledge while robots are tangible entities that

occupy physical space and whose capabilities correspond to sensors, actuators, etc. Multi-

robot systems must handle real world sensory noise, full or partial robot failures, and

communication latency or even loss of communications.

Secondly, agents are allowed to exchange resources, so the formed coalition freely

redistributes resources amongst the members. However, this is not possible in a multiple-

robot domain. Robot capabilities in handling sensors (camera, laser, sonar, or bumper) and

actuators (wheels or gripper) cannot be autonomously exchanged. This implies that a robot

coalition that simply possesses the adequate resources is not necessarily up to performing a

given task, and other locational constraints have to be represented and met in order for the

coalition to succeed.

Finally, correct resource distribution is an important issue in the robot coalition formation.

The box-pushing task (Gerkey & Mataric, 2002 b) is used to illustrate this point. Three

robots, two pushers (with one bumper and one camera) and one watcher (with one laser

range finder and one camera) cooperate to complete the task. The total resource

requirements are: two bumpers, three cameras and one laser range finder. However, this

information is incomplete, as it does not accurately represent the constraints related to

sensor locations. Correct task execution requires that the laser range finder and camera

reside on a single robot while the bumper and laser range finder reside on different robots.

Therefore each candidate coalition must be verified feasibly.

Checking the feasibility of robot coalition is a Constraint Satisfaction Problem (CSP). It is

defined by a set of variables, a set of the domain values for each variable and a set of

constraint relationships between variables, which is denoted as (V,D,C). Where V is the set

of variables {V1,…,Vn} which are resources and capabilities requirements, in box-pushing

task, V1,…,Vn are the bumper, camera and laser range finder. D is the set of the domain

values which is the sum of the available robots possessing the required resources and

capabilities, D={D1,…,Dn}, where Di is the limited domain of Vi‘s all possible values. C is the

set of constraint relationships between variables, C={C1,…,Cm}, each constraint includes a

subset of V, that is {Vi,…,Vj} and a constraint relationship R ⊆ Di×…×Dj. For the box-

pushing task, two types of constraints exist, the sensors and actuators must reside either on

the same robot or on different robots. As shown in Fig. 2, locational constraints are

represented as solid arcs (same robot) and dash arcs (different robot).

B1

C1

B2

C2

L1

C3

Fig. 2. Box-pushing task constraint graph

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4.2 The evaluation criteria of robot coalition formation

Because robots are typically unable to redistribute their resources, it is possible that the

coalition will have one or a few robots as main resource providers. This kind of coalition

tends to be heavily dependent on these members for task execution that these dominating

members become indispensable. Such coalitions should be avoided in order to improve

fault tolerance. The coalition imbalance is defined as the degree of unevenness of resource

contributions made by individual members to the coalition. The perfectly balanced

coalition is where each member contributes equally (taskvalue/n) to the task. The Balance

Coefficient (BC) quantifies the coalition imbalance level. The BC can be calculated as

follows:

12n

n

BC

taskvalue

n

γ

⎡

⎢

⎣

γγ

⎤

⎥

⎦

×××

=

?

(1)

where (γ1,γ2,...,γn) is a resource distribution with a coalition C. For the coalitions of the same

size, the higher BC, the more balanced the coalition is.

In general, larger coalitions imply that the average individual contribution and the

capability requirements from each member are lower; thus larger coalitions are more

balanced. However, larger coalitions have much more costs and therefore it is necessary to

consider coalition balance and coalition size simultaneously. The Fault Tolerance Coefficient

(FTC) metric can be used to solve this problem and it is defined as follows:

( )

f n

μ

FTCBC

δ=+

(2)

where δ+μ=1, f(n)=1-e-λn is the function of coalition size. After a particular point, increasing

n will not result in a significant increase to the function value. This means that enlarging

coalition size does not yield improved performance when the number of robots increases

beyond a threshold value. This, as one might imagine, is in accordance with a realistic robot

application.

4.3 The description of robot coalition formation problem

1.

The Ability Description of Robots

All robots in the system form a robot set R={R1,R2,…,Rn}. The ability vector of Ri is

BRi=(bi1,bi2,...,bim)T, and the ability cost vector is costRi=(costi1,costi2,...,costim)T ,where costij is the

cost of the ability bij. When bij=0, it denotes Ri without the ability bij. The cost of Ri is

m

cost b , which has m kinds of abilities.

ij ij

j 1

2.

Robot coalition is a set of robots in which robots can cooperate to complete a task. A

coalition C is the nonempty subset of R. Based on the different ability attributes of the

robots, there are different ability vectors of the coalition. For the additive capacity (such as

handling, etc.), the ability of the coalition C is as follows:

=∑

The Ability Description of Robot Coalition

i

i

CR

RC

BB

∈

=∑

(3)

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For the merger capacity (such as video distance, etc.), the ability of the coalition C is as follows:

i

i

CR

RC

BB

∈

=∪

(4)

The cost of the coalition ability is defined as follows.

The additive capacity:

( )

i

m

ij ij

R C j 1

∈

D C cost b

=

=∑ ∑

(5)

The merger capacity:

( )

i

m

∑

ij ij

j 1

=

RC

D C cost b

∈

=

∪

(6)

3.

There are K tasks, denoted by

vector:

(

t1

B b ,b ,

=

The essential condition for the coalition C to finish the task t is as follows：BC≥Bt.

4.

The Definition of Coalition’ s Income

We define a reward function which is a mapping from the set of tasks to the set of real

numbers, denoted by reward: T→R+. A cost function is defined as cost: C→R+, which is a

mapping from the set of coalitions to the set of real numbers. We consider two types of cost:

•

A coalition-inherent cost measures the inherent cost (e.g., in terms of energy

consumption or computational requirements) of using particular capabilities of the

coalition. Here the main consideration is the consumption of the robot's ability to

accomplish the tasks, including the communication between the robots in the coalition

and the cost of the coalition ability. We denote it by C_cost.

•

A task-specific cost measures cost according to task-related metrics, such as time,

distance, etc. Here we mainly consider the distance. We denote the cost of the coalition

performing the task by T_cost.

Thereby, the cost function of the coalition C performing task t is denoted as:

The Requirement Description of Task Capacity

{}

12k

Tt ,t , ,t

=

?

. The task t has the ability requirement

)

T

2m

,b

?

.

()

12

Cost C,tC_costT_cost

ωϖ=+

(7)

where

cost,

coalitions, the income of the robot coalition should be defined as:

1

ϖ and

ϖ >

2

ϖ are weighted coefficient of both the coalition-inherent cost and task-specific

0

ϖ >

. According to the differences between agent coalitions and robot

1

0

,

2

( ) ( )()

Inc CFTCrew t Cost C,t

= ×⎡−⎤

⎦

⎣

(8)

where FTC is the Fault Tolerance Coefficient, rew(t) is the reward after robots accomplish

task t.

5. Low-level coalition formation

At the low level, we employ the ant colony optimization (ACO), the particle swarm and ant

colony optimization (PSACO) and the quantum-inspired ant colony optimization (QACO)

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399

for the coalition formation. Their performance of forming robot coalition for tightly-coupled

task is compared by simulation results.

5.1 Forming robot coalition by ant colony algorithm

Put m ants on n robots at random, the probability of ant k located on the Robot i choosing

Robot j is defined as follows:

( )

t

( )

t

[]

k

ijij

k

ij

ijiu

u J

∈

1/d

α

⎤

⎦

p,

1/d

k

jJ

α

⎤ ⎡

⎦ ⎣

β

β

τ

⎡

⎣

τ

⎡

⎣

⎤

⎦

=∈

∑

(9)

where Jk is the robot set that ant k has not chosen; τij(t) is the quantity of pheromone

remaining on the line between robot i and robot j; dij (i,j=1,2,…,n) is the distance between

robot i and robot j, called communication cost; α and β control the relative weights of

pheromone and communication cost. The ant will stop seeking a route when it arrives at a

certain robot and finds that the current robot coalition can accomplish the task. When all

ants have formed their task-oriented coalitions, one loop finishes. Then each candidate

coalition is checked to verify its feasibility. Update the maximal income and the intensity of

pheromone according to the following Equation.

() ( )

t

m

∑

k

ij

ij ij

k 1

=

t1

τ ρττ+←+Δ

(10)

Here

in this loop and it is defined as:

k

ij

τ

Δ

is the increment of the familiar degree between robot i and robot j given by ant k

()

1

,

0,

k

m

∑

k

k

ij

k

Inc C

if the coalition formed by ant k includes robot i and j

C

others

τ

=

⎧

⎪⎪

⎪

⎪⎩

Δ =⎨

(11)

Inc(Ck) is the income of the coalition formed by ant k. The optimal combination of parameters α,

β and ρ in this algorithm can be determined by the experimental method. The program

termination may be controlled by a fixed evolving generation or when the evolving trend is

inconspicuous. The time complexity degree is O(NC.m.n2), NC is the number of loops.

5.2 Forming robot coalition by particle swarm and ant colony optimization

Particle Swarm Optimization (PSO) was proposed by Eberhart and Kennedy (1995).

Inspired by foraging behaviours of birds, birds are viewed as particles of swarm and their

motion is affected by their own velocity, best position of individual and population in the

past. As a result, an optimal solution can be obtained in a complex solution space.

The system is initialized with a population of random particles and then the best solution

can be found through iterations. In each time step, particles update their velocity and

position by the following formula:

()()

1012kkkkkk

v c vc pbestxcgbestx

+=+−+−

(12)

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400

11kkk

xxv

++

=+

(13)

where, pbest denotes the optimal position of single particle, gbest denotes the optimal

position of whole population,

the particle,

1. Particle Swarm and Ant Colony Optimization (PSACO)

PSO is suitable for dealing with continuous optimal problems, but for discrete optimal

problems it is difficult to express the velocity of a particle. Therefore, inspired by Genetic

Algorithms,

c0

is viewed as variation operator, while

viewed as the crossover operator of current solution with the individual optimal value and

the global optimal value respectively.

The PSACO takes an ant as a particle. Ants choose their cooperative ants based on their own

information, pbest and gbest. Then the current coalition executes crossover operations with

individual optimal coalition and global optimal coalition to form new coalition. Finally, the

new coalition executes a variation operator.

The adopted crossover strategy is to choose a random position from the second string as a

crossover point. In addition, the variation rule is constructed so as to choose a random

position, if the variation bit is -1 (the robot is not chosen), its value is set 1 (the robot is

chosen), and vice versa.

2. The PSACO Algorithm

The PSACO algorithm is described as follows:

Step 1. Initialization

Set 0

NC =

,

{1,2, , }n

=

?

. Execute ACO to form m initial coalitions and then compute the

fitness Income0 of each coalition according to Eq. (8). Treat current fitness as the individual

optimal value ptbest and treat current coalition as the individual optimal value coalition

pcbest. Then, find the global optimal value gtbest and global optimal value coalition gcbest via

ptbest.

Step 2. Put m ants on n robots randomly.

for k = 1 to m

{Initialize robot coalition consisting of robots which ants initially are

located and delete these robots from

vector

k

C

Step 3. for k = 1 to m

while（

k

Ct

{Choose a robot j according to probability

current coalition. Delete j form

coalitions. }

Step 4. for k = 1 to m

Coalition

C k formed by the K-th ant crossovers with gcbest thus produces

then

Ck crossovers with pcbest to produces

to

Ck , a new coalition

C k is formed. If

fitness 1

Income according to Eq. (8). If

Income

otherwise keep

C k as the coalition of ant k. Update the values of ptbest , pcbest , gtbest ,

gcbest .

k v is the velocity of the particle,

2 c are weight coefficients.

k x is the current position of

0c ,

1c and

kv

()()

kkkk

x gbestcx pbestc

−+−

21

is

kJ

kJ . Then calculate the capability

B

of each initial coalition.}

BB

<

）

k

ij

p

by Eq. (9) and put it into

kJ . Increase the capability vector of

0( )

'

1( )

Ck , and

'

1( )

''

1( )

1( )

C k can perform the task, compute the

10

Income

>

, the new value is accepted,

Ck . After the variation operator applied

''

1( )

1( )

0( )

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Multi-Robot Task Allocation Based on Swarm Intelligence

401

Step 5. Compute the coalition income

Step 6. Update the pheromone by Eqs. (10) & (11).

Step 7. Set 1

tt

= +

,

NC

=

<

）

{1,2,

=

?

Goto Step 2.

Step 9. Output the optimal coalition and its income.

()

k

Inc C

by Eq. (8) and save the best solution.

1

NC

+ ,0

ij

τΔ=

Step 8. if（

max

NCNC

, }

n

kJ

；

5.3 Forming robot coalition by quantum-inspired ant colony optimization

Quantum-Inspired evolutionary algorithm (QEA) was proposed by Kuk-Hyun Han

(2002). It is based on the concept and principles of quantum computing (Grover, 1994)

such as a quantum bit and superposition of states. QEA performs well even with a small

population and without premature convergence as compared to the conventional genetic

algorithm.

QEA is also characterized by the representation of the individual, the evaluation function,

and the population dynamics. However, instead of using the binary, numeric and symbolic

representation, QEA uses Q-bit as a probabilistic representation which is defined as the

smallest unit of information. A Q-bit individual is defined by a string of Q-bits. The Q-bit

individual has the advantage that can represent a linear superposition of states (binary

solutions) in search space probabilistically. Thus, the Q-bit representation has a better

characteristic of population diversity than other representations.

1. Encoding with Q-bits

A number of different representations can be used to encode the solutions onto individuals

in evolutionary computation. QEA uses a new representation, called Q-bit, for a

probabilistic representation. The representation is based on the concept of Q-bit; a Q-bit

individual as well as a string of Q-bits are defined below.

Definition 1: A Q-bit is the smallest unit of information in QEA, which is defined with a pair

of numbers (α,β) as

α

β

⎣ ⎦

⎡ ⎤

⎢ ⎥

where

and

A Q-bit may be in the ‘0’ state, in the ‘1’ state, or in a linear superposition of the two.

Definition 2: An individual Q-bit as a string of Q-bits is defined as

22

1

α

2

gives the probability that the Q-bit will be found in the ‘1’ state.

β+=

.

2

α

gives the probability that the Q-bit will be found in the ‘0’ state

β

12

12

...

...

m

m

α α

β β

⎣

α

β

⎡

⎢

⎤

⎥

⎦

where

The Q-bit representation has the advantage that it is able to represent a linear superposition

of states. If there is, for instance, a three-Q-bit system with three pairs of amplitudes such as

22

1,1,2,...,

ii

im

αβ+==

.

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402

11

1

2

22

1

2

1

3

2

2

⎡

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎦

−

Then the states of the system can be represented as

1

4

31

4

31

4

31

4

3

000001 010011 100101 110 111

4444

+−−++−−

The above result means that the probabilities to represent the states 000 , 001 , 010 , 011

, 100 , 101 , 110 , 111 are 1/16, 3/16, 1/16, 3/16, 1/16, 3/16, 1/16, and 3/16, respectively.

Therefore, the three-Q-bit system contains the information of eight states.

Evolutionary computing with Q-bit representation has a better characteristic of population

diversity than other representations, since it can represent linear superposition of states

probabilistically. Only one Q-bit individual is enough to represent eight states, but in binary

representation at least eight strings, (000), (001), (010), (011), (100), (101), (110), and (111) are

needed.

2. Quantum-Inspired Ant Colony Optimization

Wang & Li (2007) proposed a novel quantum genetic algorithm for TSP. The basic idea of

quantum-inspired ant colony optimization is to make ants which have quantum

characteristics, that is, every ant is a quantum individual and encoded by the probability of

choosing cooperative robots instead of Q-bit. The QACO is added to the corresponding

observation process and repairing process (Han, 2002).

The probability coding is defined as:

0

1

P

P

⎡

⎢

⎣

⎤

⎥

⎦

where

01

1

PP

+=

. The individual is denoted as:

0

0 1020

11

1121

j

m

jm

k

kkk

k

kk

kk

p

p

p

p

p

p

p

q

p

⎡

⎢

⎣

⎤

⎥

⎥

⎦

=⎢

(14)

where

QACO is denoted as:

of observing k-th individual,

0, it means that robot j is not chosen while the value 1 means robot j is chosen.

The algorithm of QACO is given as follows:

Step 1. Initialize

0

=

t

,

0

=

NC

,

NC

ττ=

ij

Step 2. Put m ants on n robots randomly

for 1

k =

to m

for

1j =

to n

{ if ant k starts from robot j , then

1,2,,kn

=

?

,

1,2,

( ) Q t

,

q

jm

=

?

=

,

,

t

k

1 j

,

?

{

x

k

}

k

t

ij

PP

q

t

x

=

,

.

?

0

( )

x

1

,

1

=

}

jj

k

P t

k

t

PP

= −

. The t-th generation population of

}

2

,,

n

XX

?

, where

t

kj

is either 0 or 1. When its value is

{

12

=

,

tt

n

q

{

,

1

tt

X

t

k

X is the state

12

tt

kk km

X

x

N

=

max

,

mnumAnt=

,

nnumRobot=

,

0

=Δ

ij

τ

,

0

) 0 (

1

1

jk P

=

. According to Eq. (9), calculate

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Multi-Robot Task Allocation Based on Swarm Intelligence

403

the probability of choosing cooperative robots,

Step 3. Observe the individuals of

Step 4. Check whether every state in ( )

Step 5. According to Eq. (8), calculate the income

Step 6. Save the optimization coalition b and its income

Step 7. Update the pheromone by Eqs. (10) & (11).

Step 8. Set 1tt

= +

, NCNC

=

Step 9. If ()

max

NC NC

<

and not keep evolving for a long time then go to Step 2, else

output the optimization coalition and its income.

Step 10. Repair the state which is not a solution through repairing process. If states in ( )

are all solutions, then go to Step 5.

1 j

k

k ij

PP

=

,

0

1

j

k

k ij

PP

= −

}

( )

P t is a solution, if not then go to Step10 and repair it.

()

j

Inc X of X .

( ) Inc b .

Q t and get the states ( )P t .

tt

j

1

+ , 0

ij

τ

Δ=

P t

6. High-level task allocation

The following parameters are introduced; m denotes the number of ants, each task is

denoted as node 0, and the candidate robots or robot coalition are labelled as node 1 to n.

The probability that ant k moves from node 0 to node j is formulated below:

( )

t

( )

t

[]

i

ij

τ

⎡

⎣

ij

k

ij

iu iu

u J

∈

1/cost

α

⎤

⎦

p,

1/cost

i

jJ

α

⎤ ⎡

⎦ ⎣

β

β

τ

⎡

⎣

⎤

⎦

=∈

∑

(15)

where Ji is the set of candidate robots or robot coalition to task i, and costij is the cost of

robots or robot coalition to finish task i. If the task can be completed by a single robot, the

cost is both the distance of the robot to the task and its ability consumption. Otherwise, the

cost Cost(C,t) is the cost of robot coalition to complete the task. For each ant k, the first task

node in the task list is the beginning point for the optimization. After ant k chooses an

undertaker, it moves to next task to choose an undertaker for next task, and so on. When ant

k has chosen undertakers for all tasks, one task allocation is finished. When all ants have

completed a solution, one cycle is completed. The solution with the maximal income is the

optimal solution, and then updates the intensity of pheromone according to Eq. (10).

However,

ij

τ

Δ

is defined as follows:

k

k

ij

m

∑

kj

k 1

=

Q

, Q is a constant

cost

τΔ=

(16)

The detailed task allocation algorithm is as follows:

Step 1. Initialization

Set

0, 0,(0)

ij

t NC

ττ===

requirement of each task, capability vector and cost vector of each robot.

Step 2. for i=1 to s

for k=1 to m do

{Ant k starts from the first task and determines whether the current task i is a

tightly-coupled task. If it is, then go to step 7, else choose an undertaker from

0

, 0,,,

ij

numTasks numAnt m numRobotn

τΔ====

, the capability

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Multi-Robot Systems, Trends and Development

404

iJ according to

task and repeats the above process until all tasks have been allocated to

undertakers.}

Step 3. Calculate total income of the task allocation formed by each ant. Then, update the

maximal income and the allocation schema.

Step 4. For ant k=1 to m do

Update the intensity of pheromone

ij

τ

Step 5. Set

0 , 1 , 1

=Δ+=+=

ij

NC NCtt

τ

.

Step 6. If (NC<NCmax) and (still keep evolving) then go to step 2

else output the allocation schema with the maximal income and stop the program.

Step 7. Call coalition formation algorithm ACO, PSACO and QACO to form a coalition for

task i, then goto Step 2.

The allocating process is finished by the algorithm above. If current task is tightly-

coupled, the high-level algorithm will call the low-level algorithm to form a coalition

formation.

k

ij

p by Eq. (15) and calculate the income. Then, ant k moves to next

()

1t

+

according to Eqs. (10) & (16).

7. Deadlock elimination

Because robots are fully distributed in the system with equal status among them, it is likely

to appear deadlock due to robots waiting each other at different task position. We employ a

simple strategy to avoid the deadlock. Each robot has a task queue. Robots perform tasks in

the same order as the tasks are allocated.

8. Simulation

In order to verify the effectiveness of proposed algorithms, we implement the algorithms in

the TeamBots platform developed by Carnegie Mellon University and Georgia Institute of

Technology. The implementation runs on a PC with M CPU 750, 1.8GHz Intel Pentium

processor. Based on the transportation mission, there are some tasks in the environment.

Some of them can be carried out by a single robot (loosely-coupled task) and the others must

be completed by multiple robots (tightly-coupled task). Tables 1 and 2 list the capability of

robots and task requirement.

According to Tables 1 and 2, we can find that tasks T1, T3, T6 and T9 must be completed by

multiple robots. Simulation parameters are as follows:

The high-level ant colony size m=20, low-level colony size n=20, the maximal iteration

number

max

NC 500

=

,Q1

= ,

( )

i

rew T1000

=

,

and0.9

ρ =

The task allocation algorithm was run 10 times. A comparison of three coalition formation

algorithms is given in Fig. 3 and Table 3.

From Fig. 2 and Table 3, the following conclusions can be made:

1. The effectiveness of ACO is poor and it is easy to enter into premature convergence

2. The quality of PSACO is best, however, because each ant takes longer time than other

two methods to finish a cycle, the runtime is relative long

3. QACO can find a good solution in a short time, so it is suitable for large-scale multi robots

systems

0.5

δμλ===

,

12

1

ϖϖ

== , 1.5

α =

,2

β =

,

Page 13

Multi-Robot Task Allocation Based on Swarm Intelligence

405

Robot

R0

R1

R2

R3

R4

R5

R6

R7

Capacity

1, 0, 1

1, 1, 1

2, 1, 2

1, 2, 1

0, 1, 1

1, 1, 2

0, 1, 1

2, 2, 1

Cost

1, 2, 1

1, 1, 2

2, 3, 1

3, 2, 1

1, 1, 1

2, 1, 1

3, 2, 3

3, 2, 1

Robot

R8

R9

R10

R11

R12

R13

R14

Capacity

3, 2, 1

3, 1, 1

2, 0, 1

1, 3, 3

2, 1, 3

0, 2, 1

1, 2, 3

Cost

2, 1, 3

2, 3, 2

1, 4, 3

2, 2, 1

1, 3, 1

3, 1, 2

4, 2, 1

Table 1. Capacity vector and Cost Vector of Robots

Capacity

Required

T0 1 1 1

T1 3 2 3

T2 1 2 1

T3 3 3 1

T4 2 1 1

Task

Position Task

Capacity

requirement

1 1 2

3 3 2

1 2 3

2 1 2

3 2 4

Position

(13,10)

(0,0)

(10,-10)

(-10,10)

(-8,-3)

T5

T6

T7

T8

T9

(15,0)

(0,-10)

(20,-10)

(20,20)

(20,0)

Table 2. Task requirement information

Generations

Income

Fig. 3. Optimal evolution curves

Algorithm

Best

(Generations)

9012

9030

9022

Worst

(Generations)

8953

9030

8971

Average

(Generations)

8988

9030

8997

Average

runtime（Sec）

7.37

8.52

3.75

ACO

PSACO

QACO

Table 3. Results comparison among ACO, PSACO and QACO

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Multi-Robot Systems, Trends and Development

406

9. Conclusion

This paper discusses the key issues of robot coalition formation. A task allocation

mechanism based on swarm intelligence is proposed. This allocation method adopts a

hierarchical architecture. At the high level, we employ Ant Colony Algorithm to find

optimal allocations; each ant forms a task allocation so as to choose an undertaker for every

task. At the low level, each ant forms a task-oriented robot coalition to perform a tightly-

coupled task. ACO, PSACO and QACO are used to form the coalition. The algorithm is

implemented in the TeamBots platform. Simulation results show that the proposed

approaches can effectively achieve loosely-coupled and tightly-coupled task allocation in

large-scale multi-robot systems. PSACO achieves the best solution, but its running time is

the longest. On the other hand, although QACO is somewhat inferior to PSACO in the

solution quality, its running time is only half of two other methods. Therefore, QACO is

more suitable for the large-scale multi-robot system. Our future work is to improve the

performance of algorithms and accelerate their convergence.

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