2The coordination generalized particle model—An evolutionary approach to multi-sensor fusion

2The coordination generalized particle model—An evolutionary approach to multi-sensor fusion
2The coordination generalized particle model—An evolutionary approach to multi-sensor fusion

The coordination generalized particle model—An evolutionary

approach to multi-sensor fusion

Xiang Feng

a,*

,Francis https://www.360docs.net/doc/0410065210.html,u a ,Dianxun Shuai

b

a

Department of Computer Science,The University of Hong Kong,Hong Kong

b

Department of Computer Science and Engineering,East China University of Science and Technology,Shanghai 200237,PR China

Received 9November 2006;received in revised form 6January 2007;accepted 6January 2007

Available online 14January 2007

Abstract

The rising popularity of multi-source,multi-sensor networks supporting real-life applications calls for an e?cient and intelligent approach to information fusion.Traditional optimization techniques often fail to meet the demands.The evolutionary approach pro-vides a valuable alternative due to its inherent parallel nature and its ability to deal with di?cult problems.We present a new evolution-ary approach based on the coordination generalized particle model (C-GPM)which is founded on the laws of physics.C-GPM treats sensors in the network as distributed intelligent agents with various degrees of autonomy.Existing approaches based on intelligent agents cannot completely answer the question of how their agents could coordinate their decisions in a complex environment.The proposed C-GPM approach can model the autonomy of as well as the social coordinations and interactive behaviors among sensors in a decen-tralized paradigm.Although the other existing evolutionary algorithms have their respective advantages,they may not be able to capture the entire dynamics inherent in the problem,especially those that are high-dimensional,highly nonlinear,and random.The C-GPM approach can overcome such limitations.We develop the C-GPM approach as a physics-based evolutionary approach that can describe such complex behaviors and dynamics of multiple sensors.ó2007Elsevier B.V.All rights reserved.

Keywords:Multi-sensor fusion;Sensor behavior;Sensor coordination;Evolutionary algorithm;Dynamic sensor resource allocation problem;Coordi-nation generalized particle model (C-GPM)

1.Introduction

Sensor fusion is a method of integrating signals from multiple sources into a single signal or piece of informa-tion.These sources are sensors or devices that allow for perception or measurement of the changing environment.The method uses ‘‘sensor fusion’’or ‘‘data fusion’’algo-rithms which can be classi?ed into di?erent groups,includ-ing (1)fusion based on probabilistic models,(2)fusion based on least-squares techniques,and (3)intelligent fusion.This paper presents an evolutionary approach to intelligent information fusion.

Many applications in multi-sensor information fusion can be stated as optimization problems.Among the many di?erent optimization techniques,evolutionary algorithms (EA)are a heuristic-based global search and optimization methods that have found their way into almost every area of real world optimization problems.EA provide a valu-able alternative to traditional methods because of their inherent parallelism and their ability to deal with di?cult problems that feature non-homogeneous,noisy,incom-plete and/or obscured information,constrained resources,and massive processing of large amounts of data.Tradi-tional methods based on correlation,mutual information,local optimization,and sequential processing may perform poorly.EA are inspired by the principles of natural evolu-tion and genetics.Popular EA include genetic algorithm (GA)[1],simulated annealing algorithm (SA)[2],ant

1566-2535/$-see front matter ó2007Elsevier B.V.All rights reserved.doi:10.1016/j.in?us.2007.01.001

*

Corresponding author.Tel.:+852********;fax:+852********.E-mail address:xfeng@cs.hku.hk (X.Feng).

https://www.360docs.net/doc/0410065210.html,/locate/in?us

Available online at https://www.360docs.net/doc/0410065210.html,

Information Fusion 9(2008)

450–464

colony optimization(ACO)[3],particle swarm optimiza-tion(PSO)[4],etc.,which have all been featured in either Nature or Science.

In this paper,we propose the C-GPM approach as a new branch of EA,which is based on the laws of physics. Just like other EA drawing from observations of physical processes that occur in nature,the C-GPM approach is inspired by physical models of particle dynamics.Although the other existing EA have their respective advantages,they may not be able to capture the entire dynamics inherent in the problem,especially those that are high-dimensional, highly nonlinear,and random.The C-GPM approach can overcome such limitations.We develop the C-GPM approach as a physics-based evolutionary approach that can describe the complex behaviors and dynamics arising from interactions among multiple sensors.

Our C-GPM algorithm,just like the other popular EA mentioned above,belongs to the class of meta-heuristics in arti?cial intelligence,which are approximate algorithms for obtaining good enough solutions to hard combinatorial optimization problems in a reasonable amount of compu-tation time.

Similar to the other EA,the C-GPM algorithm is inher-ently parallel and can perform well in providing approxi-mating solutions to all types of problems.EA applied to the modeling of biological evolution are generally limited to explorations of micro-evolutionary processes.Some computer simulations,such as Tierra and Avida,however, attempt to model macro-evolutionary dynamics.C-GPM algorithm is an exploration of micro-evolutionary processes.

In the physical world,mutual attraction between parti-cles causes motion.The reaction of a particle to the?eld of potential would change the particle’s coordinates and energies.The change in the state of the particle is a result of the in?uence of the potential.In C-GPM,each particle is described by some di?erential dynamic equations,and the results of their calculations govern the movement(to a new state in the?eld)of the particle.Speci?cally,each particle computes the combined e?ect of its own autono-mous self-driving force,the?eld potential and the interac-tion potential.If the particles cannot eventually reach an equilibrium,they will proceed to execute a goal-satisfaction process.

In summary,the relative di?erences between our C-GPM algorithm and other popular EA can be seen in Table1.The common features of these di?erent approaches are listed as follows:

?They draw from observations of physical processes that occur in nature.

?They belong to the class of meta-heuristics,which are approximate algorithms used to obtain good enough solutions to hard combinatorial optimization problems in a reasonable amount of computation time.

?They have inherent parallelism and the ability to deal with di?cult problems.

?They consistently perform well in?nding approximating solutions to all types of problems.

?They are mainly used in the?elds of arti?cial intelligence.

In this paper,we study some theoretical foundations of the C-GPM approach including the convergence of C-GPM.The structure of the paper is as follows.In Section 2,we discuss and formalize the problem model for the typ-ical multi-sensor system.In Section3,we present the evo-lutionary C-GPM approach to intelligent multi-sensor information fusion.In Section4,we describe an experiment to verify the claimed properties of the approach.We draw conclusion in Section5.

2.Dynamic sensor resource allocation

In a sensor-based application with command and con-trol,a major prerequisite to the success of the command and control process is the e?ective use of the scarce and costly sensing resources.These resources represent an important source of information on which the command and control process bases most of its reasoning.Whenever there are insu?cient resources to perform all the desired tasks,the sensor management must allocate the available sensors to those tasks that could maximize the e?ectiveness of the sensing process.

Table1

The C-GPM algorithm vs.other popular EA

C-GPM GA SA ACO PSO

Inspired by Physical models of particle

dynamics Natural

evolution

Thermodynamics Behaviors of

real ants

Biological swarm

(e.g.,swarm of bees)

Key components Energy function;di?erential

dynamic equations Chromosomes Energy function Pheromone laid Velocity-coordinate

model

Exploration Both macro-evolutionary and

micro-evolutionary processes Macro-

evolutionary

processes

Micro-

evolutionary

processes

Macro-

evolutionary

processes

Macro-evolutionary

Dynamics Can capture the entire dynamics

inherent in the problem Cannot capture Can capture

partly

Cannot capture Cannot capture

High-dimensional,highly nonlinear, random behaviors and dynamics Can describe Cannot describe Can describe

partly

Cannot describe Cannot describe X.Feng et al./Information Fusion9(2008)450–464451

The dynamic sensor allocation problem consists of selecting sensors of a multi-sensor system to be applied to various objects of interest using feedback strategies.Con-sider the problem of n sensors,A ?f A 1;...;A n g ,and m objects,T ?f T 1;...;T m g .In order to obtain useful infor-mation about the state of each object,appropriate sensors should be assigned to various objects at the time intervals t 2f 0;1;...;T à1g .The collection of sensors applied to object k during interval t is represented by a vector X k et T?f x 1k ;...;x ik ;...;x nk g ,where

x ik et T?

1if sensor i is used on object k at interval t

0otherwise &

Because of the limited resources sustaining the whole system,the planned sensor distributions must satisfy the following constraint for every t 2f 0;1;...;T à1g :

X m k ?1

r ik et Tx ik et T?1e1T

where r ik denotes that quantity of resources consumed by sensor i on object k and 06r ik 61.

The goal of sensor allocation is to try to achieve an opti-mal allocation of all sensors to all the objects after T stages.Let C ?ec ik Tn ?m be a two-dimensional weight vector.Sensor allocation can be de?ned as a problem to ?nd a two-dimensional allocation vector R ?er ik Tn ?m ,which max-imizes the objective in (2),subject to the constraint (1)z eR T?eC TT

RX ?

X n i ?1

X m k ?1

c ik r ik x ik e2T

Let f ik et Trepresent the intention strength of social

coordination.Thus we obtain an allocation-related matrix S et T??s ik et T n ?m ,as shown in Table 2,where s ik et T?h r ik et T;c ik et T;x ik et T;f ik et Ti :For convenience,both r ik et Tand c ik et Tare normalized such that 06r ik et T61and 06c ik et T61.

3.The C-GPM approach to sensor fusion 3.1.Physical model of C-GPM

This subsection discusses the physical meanings of the coordination generalized particle model (C-GPM)for sensor fusion in multi-sensor systems which involve social coordi-nations among the sensors.C-GPM treats every entry of

the allocation-related matrix S ik as a particle,s ik ,in a force ?eld.The problem solving process is hence transformed into one dealing with the kinematics and dynamics of par-ticles in the force ?eld.The s ik ’s form the matrix S ik .For convenience,we let s ik represent both an entry of the matrix as well as its corresponding particle in the force ?eld.

Particle and force-?eld are two concepts of physics.Par-ticles in our C-GPM move not only under outside forces,but also under internal force;hence in this sense,they are somewhat di?erent from particles in physics.

As shown in Fig.1,the vertical coordinate of a particle s ik in force ?eld F represents the utility obtained by sensor A i from being used on object T k .A particle experiences several kinds of forces simultaneously,which include the gravitational force of the force ?eld,the pulling or pushing forces due to social coordinations among the sensors,and the particle’s own autonomous driving force.The forces on a particle are handled only along the vertical direction.Thus a particle will be driven by the resultant force of all the forces that act on it upwards or downwards along the vertical direction.The larger the upward/downward resul-tant force on a particle,the faster the upward/downward motion of the particle.When the resultant force on a par-ticle is equal to zero,the particle will stop moving,being in an equilibrium state.

The upward gravitational force of the force ?eld con-tributes an upward component of a particle’s motion,which represents the tendency that the particle pursues the collective bene?t of the whole multi-sensor system.The other upward or downward components of the parti-cle’s motion,which are related to the social coordinations among the sensors,depend on the strengths and kinds of these coordinations.The pulling or pushing forces among particles make particles move to satisfy resource restric-tions,as well as re?ect the social coordinations and behav-iors among the sensors.A particle’s own autonomous driving force is directly proportional to the degree the par-ticle tries to maximize its own pro?t (utility).This autono-mous driving force of a particle actually sets the C-GPM approach apart from the classical physical model.All the generalized particles simultaneously move in the force ?eld,and once they have all reached their respective equilibrium positions,we have a feasible solution to the optimization problem in question.

Table 2

The matrix S et TSensors Objects T 1

...T k

...T m

A 1r 11;c 11;x 11;f 11...r 1k ;c 1k ;x 1k ;f 1k ...r 1m ;c 1m ;x 1m ;f 1m ............A i r i 1;c i 1;x i 1;f i 1...r ik ;c ik ;x ik ;f ik ...r im ;c im ;x im ;f im .....

...

...

.A n

r n 1;c n 1;x n 1;f n 1

...

r nk ;c nk ;x nk ;f nk

...

r nm ;c nm ;x nm ;f

nm

452

X.Feng et al./Information Fusion 9(2008)450–464

Because the problem in this paper is a one objective problem,we limit the particles movements to one dimen-sion.The design of C-GPM in fact allows forces of all directions to exist.These forces can be decomposed into their horizontal and vertical components.In this present work,only the vertical component may a?ect a particle’s motion.In a forthcoming paper,we will introduce the mul-tiple objectives problem where we will handle particles’movements along multiple dimensions.3.2.Mathematical model of C-GPM

We de?ne in this subsection the mathematical model of C-GPM for the sensor allocation problem that involves n sensors and m objects.

Let u ik et Tbe the distance from the current position of particle s ik to the bottom boundary of force ?eld F at time t ,and let J et Tbe the utility sum of all particles,which we de?ne as follows:

u ik et T?a ?1àexp eàc ik et Tr ik et Tx ik et TT

J et T?X

n i ?1

X m k ?1

u ik et T

e3T

where 0

At time t ,the potential energy functions,P et T,which is caused by the upward gravitational force of force ?eld F ,is de?ned by

P et T? 2ln X n i ?1

X m k ?1

exp ?àu 2ik et T=2 2 à 2

ln mn

e4Twhere 0< <1.The smaller P et Tis,the better.With Eq.(4),we attempt to construct a potential energy function,P et T,such that the decrease of its value would imply the in-crease of the minimal utility of all the sensors.We prove it in Proposition 3.This way we can optimize the multi-sen-sor fusion problem in the sense that we consider not only the aggregate utility,but also the individual personal utili-ties,especially the minimum one.In addition, represents the strength of upward gravitational force of the force ?eld.The bigger is,the better.If we did not get a su?ciently satisfactory result by C-GPM,we can let smaller.

The gravitational force of the force ?eld causes the par-ticles to move to increase the corresponding sensors’mini-mal personal utility,and hence to realize max–min fair allocation and increase the whole utility of the multi-sensor system.

Following the literature [5–8],we divide typical social coordinations between sensor A i and sensor A j into 12pos-sible types,as in Table 3.

A ijk :To avoid the harmful consequence possibly caused by A j ,A i wants to change its own current intention.

B ijk :To exploit the bene?cial consequence possibly caused by A j ,A i wants to change its own current intention.

C ijk :To bene?t A j ,A i wants to change its own current intention regardless of self-bene?t.

D ijk :A i tries to allure A j to modify A j ’s cur-rent intention so that A i could avoid the harmful conse-quence possibly caused by A j .

E ijk :A i tries to entice A j to modify A j ’s current intention so that A i could exploit the bene?cial consequence possibly caused by A j .

F ijk :A i tries to tempt A j to modify A j ’s current intention so that A i might bene?t from this,while A j ’s interests might be infringed.

G ijk :To compete with each other,neither A i nor A j will modify their own intention,but both A i and A j might enhance their intention strengths with respect to the K th goal (or object).

H ijk :Neither A i nor A j will modify their own current intention,but both A i and A j might decrease their intention strengths with respect to the K th goal.

I ijk :Due to disregard of the other side,neither A i nor A j will modify their own current intention.

J ijk :To harm the other side,both A i and A j try to modify their own current intentions.

K ijk :Both A i and A j try to modify their own current intentions so that they could implement the intention of the other side.

L ijk :Both A i and A j try to modify their current intentions so that they can do some-thing else.

Of the 12types of social coordination,types A ,B ,C ,D ,E and F and F are via unilateral communication,and types G,H,I,J,K and L by bilateral communication.Based on which sensor(s)will modify their current intention,the 12types can be conveniently grouped into the four categories.For A ijk ;B ijk ;C ijk ,it is A i ;for D ijk ;E ijk ;F ijk ,it is A j ;

for

Fig.2.Graphical presentation of u ik et T.

Table 3

Social coordinations among sensors b ijk Type Name

f ijk I

A ijk Adaptive avoidance coordination à1

B ijk Adaptive exploitation coordination

C ijk Collaboration coordination II

D ijk Tempting avoidance coordination 1

E ijk Tempting exploitation coordination

F ijk Deception coordination

III G ijk Competition coordination 1

H ijk Coalition coordination

I ijk Habituation/preference coordination IV J ijk Antagonism coordination à1

K ijk Reciprocation coordination L ijk

Compromise coordination

X.Feng et al./Information Fusion 9(2008)450–464453

G ijk;H ijk;I ijk,none will;and for J ijk;K ijk;L ijk,both A i and A j will,and so we have

LeIT?Le10T?f A ijk;B ijk;C ijk j8i;j;k g;

LeIIT?Le01T?f D ijk;E ijk;F ijk j8i;j;k g;

LeIIIT?Le00T?f G ijk;H ijk;I ijk j8i;j;k g;

LeIVT?Le11T?f J ijk;K ijk;L ijk j8i;j;k g;

L?eLe01T[Le10T[Le10T[Le11TT:

The intention strength f iketTof sensor A i with respect to object T k is de?ned by

f iketT?

X n

j?1f ijketTt

X n

j?1

f jiketTe5T

f ijketT?

1if b ijk2LeIIT[LeIIITà1if b ijk2LeIT[LeIVT(

f jiketT?

1if b jik2LeIT[LeIIIT

à1if b jik2LeIIT[LeIVT

(

e6T

b ijk is the social coordination of sensor A i with respect to sensor A j for object T k,which gives rise to the change f ijketTof intention strength f iketT.

f iketTof s iketTrepresents the aggregate intention strength when more than one social coordination happen simulta-neously at time t.The greater f iketTis,the more necessary would sensor A i have to modify its r iketTfor object T k.

At time t,the potential energy function,QetT,is de?ned by

QetT?

X n

i?1X m

k?1

r iketTx iketTà1

2

à

X

i;k Z u ik

f?1texpeàf ik xT à1à0:5g d xe7T

The?rst term of QetTis related to the constraints on the sensors’capability;the second term involves social coordi-nations among the sensors,with f ik coming from Eqs.(5) and(6).The?rst term of QetTcorresponds to a penalty function with respect to the constraint on the utilization of resources.Therefore,the sensors’resource utilization can be explicitly included as optimization objectives in the multi-sensor fusion problem.The second term of QetTis chosen as shown because we want o Q

ik

to be a monotone decreasing sigmoid function,as shown in Fig.3.àf?1texpeàf ik u ikT à1à0:5g is such a function.Therefore

we let o Q

o u ik equal toàf?1texpeàf ik u ikT à1à0:5g.Then o Q

o u ik

is

integrated to be Q.

A particle in the force?eld can move upward along a vertical line under a composite force making up of

?the upward gravitational force of the force?eld,

?the upward or downward component of particle motion that is related to social coordinations among the sensors,?the pulling or pushing forces among the particles in order to satisfy resource restrictions,and

?the particle’s own autonomous driving force.

The four kinds of forces can all contribute to the parti-cles’upward movements.What is more,these forces pro-duce hybrid potential energy of the force?eld.The general hybrid potential energy function for particle s ik, E iketT,can be de?ned by

E iketT?k1u iketTtk2JetTàk3PetTàk4QetTe8Twhere0

Dynamic equations for particle s ik is de?ned by

d u iketT=d t?W1etTtW2etT

W1etT?àu iketTtc v iketT

W2etT?k1tk2o JetT

ik

àk3o PetT

ik

àk4o QetT

ik

h i

o u iketT

ik

h i2

to u iketT

ik

h i2

&'

8

>>>

>>>

<

>>>

>>>

:

e9T

where c>1.And v iketTis a piecewise linear function of u iketTde?ned by

v iketT?

0if u iketT<0

u iketTif06u iketT61

1if u iketT>1

8

><

>:e10T

In order to dynamically optimize sensor allocation,the particle s ik may alternately modify r ik and c ik,respectively, as follows:

d c iketT=d t?k1

o u iketT

o c iketT

tk2

o JetT

o c iketT

àk3

o PetT

o c iketT

àk4

o QetT

o c iketT

e11Td r iketT=d t?k1

o u iketT

o r iketT

tk2

o JetT

o r iketT

àk3

o PetT

o r iketT

àk4

o QetT

o r iketT

e12T

where o QetT

ik

?àf?1texpeàf iketTu iketTT à1à0:5g is a sig-moid function of the aggregate intention strength f iketT. The graphical presentation of o Q

ik

is shown in Fig.3.

Note that f iketTis related to social coordinations among

the sensors at time t.o Q

o u ik

is a monotone decreasing function. By Eq.(9)the greater the value of f iketT,the more there

are Fig.3.Graphical presentation of o Q

o u ik

.

454X.Feng et al./Information Fusion9(2008)450–464

values of W 2et Tand hence u ik et Twill increase,which implies that the social coordinations will strengthen the current allocation r ik et T.Because o Q o u ik

is a monotone decreasing function,the greater the value of f ik et T,the smaller o Q ik ,the greater ào Q ik

,the greater W 2et Tby Eq.(9)and the greater D r ik et t1Tby Eq.(12).Since r ik et t1T?r ik et TtD r ik et t1T,the greater the value of f ik et T,the greater r ik et t1T.Since u ik is a monotone increasing function,the greater r ik et t1T,the greater u ik f ik et T")

o Q o u ik #)ào Q o u ik ")e9TW 2"ào Q o u ik

")e11T

D r ik et t1T")a

r ik et t1T")e3Tb

u ik "

(a)r ik et t1T?r ik et TtD r ik et t1T;

(b)u ik is monotone increasing function.

In the following,we derive some formal properties of the mathematical model presented above.

Proposition 1.Updating the weights c ik and allotted resource r ik by Eqs.(11)and (12)respectively amounts to changing the speed of particle s ik by W 2et Tof Eq.(9).

Denote the j th terms of Eqs.(11)and (12)by d c ik et T

D E j and d r ik et T

D E j

;respectively.When allotted resource r ik is updated according to (12),the ?rst and second terms of (12)will cause the following speed increments of the parti-cle s ik ,respectively:

h d u ik et T=d t i r

1?

o u ik et To r ik et Td r ik et Td t ()1

?k 1o u ik et To r ik et T !2

e13Th d u ik et T=d t i r

2?

o u ik et To r ik et Td r ik et Td t ()2

?k 2o u ik et To r ik et To J et To r ik et T?k 2

o u ik et To r ik et To J et To u ik et To u ik et To r ik et T

?k 2

o J et To u ik et To u ik et T

o r ik et T !2

e14TSimilarly,the third and the fourth term of Eq.(12)will cause the following speed increments of the particle s ik :

h d u ik et T=d t i r 3?àk 3o P et Tik o u ik et Tik !2

h d u ik et T=d t i r

4

?àk 4

o Q et To u ik et To u ik et T

o r ik et T

!2

Similarly,for Eq.(11),we have h d u ik et T=d t i c

j ;j ?1;2;3;4.We thus obtain

X 4j ?1

?h d u ik et T=d t i c j th d u ik et T=d t i r j

?k 1tk 2

o J et To u ik et Tàk 3o P et To u ik et Tàk 4o Q et T

o u ik et T

!o u ik et To r ik et T

!2

(to u ik et To c ik et T

!2

)?W 2et T:Therefore,updating c ej T

ik and r ej T

ik by (11)and (12),respec-tively,gives rise to the speed increment of particle s ik that is exactly equal to W 2et Tof Eq.(9).

Proposition 2.The ?rst and second terms of Eqs.(11)and (12)will enable the particle s ik to move upwards,that is,the personal utility of sensor A i from object T k increases,in direct proportion to the value of ek 1tk 2T.

According to Eqs.(13)and (14),the sum of the ?rst and second terms of Eqs.(11)and (12)will be

h d u ik et T=d t i r 1th d u ik et T=d t i r 2th d u ik et T=d t i c 1th d u ik et T=d t i c

2

?k 1tk 2o J et To u ik et T !o u ik et To r ik et T !2t

o u ik et T

o c ik et T !2()

?ek 1tk 2Tx 2ik et T?r 2ik et Ttc 2

ik et T ?àu ik et T 2P 0:

Therefore,the ?rst and second terms of (11)and (12)will

cause u ik et Tto monotonically increase.

Proposition 3.For C-GPM,if is very small,then decreas-ing the potential energy P et Tof Eq.(4)amounts to increasing the minimal utility of an sensor with respect to an object,min-imized over S et T.

Supposing that H et T?max i ;k fàu 2ik et Tg ,we have

?exp eH et T=2 2T 2 2

6X n i ?1

X m k ?1

exp eàu 2ik et T=2 2

T

"#2 2

6?mn exp eH et T=2 2T 2 2

:

Taking the logarithm of both sides of the above inequalities gives

H et T62 2ln X n i ?1

X m k ?1

exp eàu 2ik et T=2 2T6H et Tt2 2

ln mn :

Since mn is constant and is very small,we have

H et T%2 2ln X n i ?1

X m k ?1

exp eàu 2ik et T=2 2Tà2 2

ln mn ?2P et T:

It turns out that the potential energy P et Tat the time t rep-resents the maximum of àu 2ik et Tamong all the particles s ik ,which is the minimal personal utility of a sensor with respect to an object at time t .Hence the decrease of potential energy P et Twill result in the increase of the minimum of u ik et T.Proposition 4.Updating c ik and r ik according to Eqs.(11)and (12)amounts to increasing the minimal utility of a sensor with respect to an object in direct proportion to the value of k 3.

The speed increment of particle s ik ,which is related to potential energy P et T,is given by d u ik et Td t ()3

?h d u ik et T=d t i r 3th d u ik et T=d t i c

3

?àk 3o P et To u ik et To u ik et To r ik et T !2to u ik et T

o c ik et T

!2():

X.Feng et al./Information Fusion 9(2008)450–464455

Denote by

d P et Td t

D

E

the di?erentiation of the potential

energy function P et Twith respect to time t arising from using (11)and (12).We have d P et Td t ()?o P et To u ik et Td u ik et T

d t ()3

?àk 3o P et To u ik et T !2o u ik et To r ik et T !2t

o u ik et T

o c ik et T

!2()?àk 3x 2ik et Tu 2ik et Tx 2ik et T?r 2ik et Ttc 2

ik et T ?u ik et T 2

60:

where,x ik et T?

exp ?àu 2ik et T=2 2

X n i ?1

X m k ?1

exp ?àu 2ik et T=2 2

:

,

It can be seen that using Eqs.(11)and (12)gives rise to

monotonic increase of P et T.Then by Proposition 3,the de-crease of P et Twill result in the increase of the minimal util-ity,in direct proportion to the value of k 3.

Proposition 5.Updating c ik and r ik by Eqs.(11)and (12)gives rise to monotonic increase of the whole utility of all the sensors,in direct proportion to the value of k 2.

Similar to Proposition 2,it follows that when a particle s ik modi?es its c ik and r ik by Eqs.(11)and (12),di?eren-tiation of J et Twith respect to time t will not be negative—i.e.,d J R et T

d t D E

P 0,and it is directly proportional to the value of k 2.

Proposition 6.Updating c ik and r ik by Eqs.(11)and (12)gives rise to monotonic decrease of the potential energy Q et T,in direct proportion to the value of k 4.As in the above,we have

d u ik et Td t ()4?àk 4o Q et To u ik et To u ik et To r ik et T !2to u ik et T

o c ik et T !2()

;and

d Q et Td t ()?

o Q et To u ik et Td u ik et Td t ()4

?àk 4o Q et T

o u ik et T !2?o u ik et To r ik et T !2to u ik et To c ik et T !2()60:

Proposition 7.C-GPM can dynamically optimize in parallel

sensor resource allocation for a collection of sensors exhibit-ing different autonomous strengths of pursuing personal util-ity and executing social coordinations.

In summary,by Propositions 1–6,ek 1tk 2Trepresents the autonomous strength for A i to pursue its own personal utility;k 2represents the autonomous strength for A i to take into account the collective utility of all the sensors;k 3rep-resents the autonomous strength for A i to increase the min-imal personal utility among all the sensors;and k 4represents the autonomous strength for constraint satisfac-tion and social coordinations.

Propositions 1,2,3,47show that di?erent sensors may have di?erent degrees of autonomy and their own rational-

ity in pursuing individual and system-wide bene?ts.More-over,di?erent sensors may engage in di?erent social

coordinations with other sensors for a speci?c object.A large variety of social coordinations may be taken into account,including possibly unilateral and unaware social coordinations.

3.3.Convergence analysis

In this subsection,we show that all the particles can con-verge to their stable equilibrium states through algorithm C-GPM.

Lemma 1.If c à1>àW 2et T>0,o W 2et T

o u ik et T<1for u ik et T>1,

and o W 2et T

o u ik et T>1àc for 01;v ik et T?1.

Proof.For c >1,W 1et Tof particle s ik is a piecewise linear function of the stimulus u ik et T,as shown in Fig.4:Segment I,Segment II,and Segment III.By Eq.(9),a point is an equilibrium point,i.e.,d u ik et T=d t ?0,i?àW 2et T?W 1et Tat the point.We see that for the case of c à1>àW 2et T>0,an equilibrium point may be on Segment I,II or III.Note from Eq.(3),u ik et TP 0.Thus we need not con-sider the equilibrium point on Segment I.

Suppose that the particle s ik is at an equilibrium point on Segment III at time t 0,and an arbitrarily small perturbation D u ik to the equilibrium point occurs at time

t 1.Since o W 2et To u ik et T<1,and o W 1et T

o u ik et T?à1for u ik et T>1,we have

c ?

o W 1et Tik to W 2et T

ik h i <0,and D

d u ik d t ?d u ik d t t 1àd u ik d t t 0?d u ik d t

t 1

?D ?W 1et TtW 2et T %o W 1et To u ik et Tto W 2et To u ik et T

!D u ik ?àj c j D u ik :

That means d u ik d t t 1

is always against D u ik ,or in other words,the perturbation will be suppressed and hence the particle s ik will return to the original equilibrium

point.

456X.Feng et al./Information Fusion 9(2008)450–464

Whereas,for an equilibrium point on Segment II,

because o W2etT

o u iketT>1àc and o W1etT

o u iketT

?cà1>0,we have

c?

o W2etT

ik t

o W1etT

ik

!

>0;and

d u ik

t1

%o W1etT

o u iketT

t

o W2etT

o u iketT

!

D u ik?j c j D u ik

such that the perturbation is intensi?ed and the particle s ik departs from the original equilibrium point on Segment II. Therefore,an equilibrium point on Segment II is unstable, and only an equilibrium point on Segment III,e.g.,p4in Fig.4,is stable with u iketT>1;v iketT?1.h

Lemma 2.If c>1;àW2etT<0,and o W2etT

o u iketT<1for

u iketT>1,then the particle s ik will converge to a stable equi-librium point with u iketT>1;v iketT?1.

Proof.Due to c>1andàW2etT<0,an equilibrium point must be on Segment III,e.g.,p6in Fig.4.Moreover,as sta-

ted in the proof of Lemma1,o W2etT

o u iketT<1for u iketT>1guar-

antees any equilibrium point on Segment III to be stable, with u iketT>1;v iketT?1.h

Lemma 3.If c>1,o W2etT

ik <1for u iketT?1t;and

o W2etTo u iketT>1àc for u iketT?0tand u iketT?1à,then the equilib-

rium points s1and s3in Fig.4are unstable and a saddle point, respectively.

Proof.It is straightforward from the proof of Lemma 1.h

Theorem 1.If c>1,o W2etT

ik

<1for u iketTP1t,and o W2etT

o u iketT

>1àc for0t6u iketT61à,the dynamical equation (9)has a stable equilibrium point on Segment III iff the right side of the equation is larger than0for u iketT?1and v iketT?1:

Proof.By Lemmas1–3,the equilibrium points on Segment II and Segment III,except for the saddle point s3in Fig.4, are unstable and stable,respectively.We denote the right side of Eq.(9)by RHS.

Suf?ciency.Assume that RHS?d u ijetT?W1etTtW2etT>0holds for u iketT?1;v iketT?1.It follows that W1etT?àW2etTfor u iketT?1;v iketT?1,namely,it is impossible that the equilibrium point is the intersection

point s3where W1etT?àW2etT.Since RHS?d u ik

d t >0at

point u iketT?1,v iketT?1,the increase of u iketTfrom value 1leads to that the particle s ik converges to a stable equilibrium point on Segment III.

Necessity.Suppose that Eq.(9)has a stable equilibrium point.We need to prove that RHS>0holds for u iketT?1 and v iketT?1.By contrary,if there is RHS60for u iketT?1and v iketT?1,then the equilibrium point must be either at the point s3where RHS?0,or on Segment II

because RHS?d u ik

d t <0would giv

e rise to the decrease of

u iketTfrom value1.Since the point s3and any equilibrium

point on Segment II are all non-stable,we have a

contradiction.h

Theorem 2.If c>1,o W2etT

ik

<1for u iketTP1t;and

o W2etT

ik

>1àc for0t6u iketT61à,then the dynamical equa-

tion(8)has a stable equilibrium point iff

c>1t2?k3t0:25k4 :e15T

Proof.By Eqs.(3),(4),(7),(8)we have

W2etT?f k1tk2àk3x2

ik

etTu2

ik

etT

àk4?e1texpeàf iketTu iketTTTà1à0:5 2g?r2

ik

etT

tc2

ik

etT ?u iketT 2e16T

Note that x2

i

etT;r2

ik

etT;c2

ik

etT61.By Theorem1,for

u iketT?1;v iketT?1,we have

c>1àW2etT

?1tfàk1àk2tk3x2

ik

etTu2

ik

etT

tk4?e1texpeàf iketTu iketTTTà1à0:5 2g?r2

ik

etTtc2

ik

etT ?u iketT 2

61t2?k

3

t0:25k4 :?

Lemma4.If c>1;f iketTP0,and the following conditions

are valid:

k1tk2

c>max f1t4?k1tk2t0:25k3tf iketTk4 ;1t2?k3t0:25k4 g:

e18T

then the dynamical equation(9)has a stable equilibrium point

with u ik?1;v ik?1.

https://www.360docs.net/doc/0410065210.html,ing Eq.(16)and noting d x iketT

ik

?2x iketT

u iketTex iketTà1T,we obtain

o W2etT

o u iketT

?f2?àk1àk2tk3x2

ik

etTu2

ik

etT

tk4?e1texpeàf iketTu iketTTTà1à0:5 2

tà2k3x iketT

d x iketT

d u iketT

u2

ik

etTà2k3x2

ik

etTu iketT

à2k4f iketTexpeàf iketTu iketTT

??e1texpeàf iketTu iketTTTà1à0:5

??1texpeàf iketTu iketTT à2

i

eàu iketTTg x2

ik

etT?r2

ik

etT

tc2

ik

etT ?àu iketT ?f2?àk1àk2tk3x2

ik

etTu2

ik

etT

tk4?e1texpeàf iketTu iketTTTà1à0:5 2

t?à4k3x2

ik

etTu3

ik

etTex iketTà1Tà2k3x2

ik

etTu iketT

à2k4f iketTexpeàf iketTu iketTT

??e1texpeàf iketTu iketTTTà1à0:5

??1texpeàf iketTu iketTT à2 eàu iketTTg x2

ik

etT

??r2

ik

etTtc2

ik

etT ?àu iketT

X.Feng et al./Information Fusion9(2008)450–464457

Then from o W2etT

o u iketT

<1for u iketTP1t;and u iketT6x iketT; r iketT;c iketT61,we derive

o W2etT

o u iketT

<2?2eàk1àk2tk3t0:25k4T

t4k3e2=3T2e3=4T3=12 <1;

which leads to Eq.(17).

Similarly,from o W2etT

o u iketT>1àc for0t6u iketT61à,we

have

o W2etT

o u iketT

>4eàk1àk2àk3=4àf iketTk4T>1àc;

which leads to Eq.(18).By Theorem2,therefore the con-clusion is valid.h

Lemma5.If c>1;f iketT60,and the following conditions are valid:

k1tk2max f1t4?k1tk2t0:25k3 ;1t2k3t0:25k4ge20Tthen the dynamical equation(8)has a stable equilibrium point with u ik?1;v ik?1:

Proof.Similar to the proof of Lemma4,from o W2etT

o u iketT<1

for u iketTP1t,and u iketT6;x iketT;r iketT;c iketT61,we derive

o W2etT

o u iketT

<2?2eàk1àk2tk3t0:25k4T

t4k3e2=3T2e3=4T31=12tj f iketTj k4 <1;

which leads to Eq.(19).

And from o W2etT

o u iketT

>1àc for0t6u iketT61à,we have

o W2etT

o u iketT

>4eàk1àk2àk3=4T>1àc;

which leads to Eq.(20).By Theorem2,therefore the con-clusion is valid.h

Theorem3.If c>1,and the conditions(18),(19)are valid, then the dynamical equation(9)has a stable equilibrium point with u ik?1;v ik?1.

Proof.Straightforward,by Lemmas4and5.h

Theorem4.If the conditions(17),(18)remain valid,then C-GPM will converge to a stable equilibrium state.

https://www.360docs.net/doc/0410065210.html,ing Eq.(16)and noting d x iketT

ik ?2x iketTu iketT

ex iketTà1T,we obtain o W2etT

o u iketT

?2?àk1àk2tk3x2

ik

etTu2

ik

etT

è

tk4?e1texpeàf iketTu iketTTTà1à0:5 2

t?à2k3x iketT

d x iketT

d u iketT

u2

ik

etTà2k3x2

ik

etTu iketT

à2k4f iketTexpeàf iketTu iketTT

??e1texpeàf iketTu iketTTTà1à0:5

?1texpeàf iketTu iketTT à2

h i

eàu iketTT

o

x2

ik

etT?r2

ik

etT

tc2

ik

etT ?àu iketT ?f2?àk1àk2tk3x2

ik

etTu2

ik

etTtk4?e1texpeàf iketTu iketTTTà1à0:5 2

t?à4k3x2

ik

etTu3

ik

etTex iketTà1Tà2k3x2

ik

etTu iketTà2k4f iketTexpeàf iketTu iketTT

??e1texpeàf iketTu iketTTTà1à0:5

??1texpeàf iketTu iketTT à2 eàu iketTTg x2

ik

etT?r2

ik

etT

tc2

ik

etT ?àu iketT

For the force?eld F,we de?ne a Lyapunov function LetTby

LetT?à

1

2

X

i;k

ecà1Tv iketT2t

X

i;k

Z t

d v ikexT

d x

fàk1àk2

tk3x2

ik

exTu2

ik

exTtk4?e1texpeàf ikexTu ikexTTTà1

à0:5 2g?r2

ik

exTtc2

ik

exT ?àu ikexT 2d x:

We hence have

j LetTj6

X

i;k

ecà1Tj v iketT2jt

X

i;k

Z t

d v ikexT

d x

ájfàk1àk2

tk3x2

ik

exTu2

ik

exTtk4?e1texpeàf ikexTu ikexTTTà1

à0:5 2gj?r2

ik

exTtc2

ik

exT ?àu ikexT 2d x:

Since condition(18)is valid,v iketT61and u ietT61,it fol-lows that

j LetTj6

X

i;k

ecà1Tt

X

i;k

c

which implies that LetTis bounded.

In addition,we have

d LetT

d t

X

i;k

ecà1Tv iketT

d v iketT

d t

t

X

i;k

d v iketT

d t

fàk1àk2

tk3x2

ik

etTu2

ik

etTtk4?e1texpeàf iketTu iketTTTà1

à0:5 2g?r2

ik

etTtc2

ik

etT ?àu iketT 2

X

i;k

d v iketT

d u ietT

d u iketT

d t

fàu iketTtc v iketTtf k1tk2

àk3x2

ik

etTu2

ik

etTàk4?e1texpeàf iketTu iketTTTà1

à0:5 2g?r2

ik

etTtc2

ik

etT ?àu iketT 2g

X

i;k

d v iketT

d u iketT

d u iketT

d t

2

60:

458X.Feng et al./Information Fusion9(2008)450–464

Thus LetTwill monotonically decrease with the elapsing time.h

3.4.The parallel C-GPM algorithm

Algorithm C-GPM

As indicated by Theorems1–4,as long as we properly select the parameters k1;k2;k3;k4for the dynamical equa-tions of C-GPM according to Eqs.(17),(19),the conver-gence and stability of the particle dynamics can be guaranteed.

Given the results above,we can construct an algorithm to solve the sensor fusion(resource allocation)problem,as given below.

The algorithm C-GPM has in general a complexity of X(nm),where(nm)is the number of particles(the sum of the number of elements of matrix S).The more realistic time complexity of the algorithm is OeIT,where I is the number of iterations for Step2(the while loop),as much of the inner steps can be carried out in parallel.

4.Simulations

Here we give an example of a robot which represents a multi-sensor fusion problem,and then use our C-GPM method to?nd the optimum solution.

The robot has seven sensors:eye,nose,ear,arm,hand, leg and foot.The robot has a reportoire of three actions: open a door,take dictation,?nd food and take back. Now we describe in detail how to solve the multi-sensor fusion problem,namely,how to allocate sensors to objects.

Let sensors A i;i?1;2;...;7represent the eye,nose, ear,arm,hand,leg and foot of the robot,respectively. Let objects T k;k?1;2;3represent the robot’s actions of opening a door,taking dictation,and?nding food and tak-ing them back,respectively.

Step1.

1.x ik:

First,we establish the relations between sensors and objects(actions).If sensor A i is related to object T k, x ik?1;otherwise,x ik?0.For example,the eye,arm,hand,leg and foot are related to the object of opening a door;the eye,ear,arm and hand are related to the object of taking dictation.We can form the matrix X,as follows:

2.c ik:

Based on the di?erent e?ects of the sensors A1–A7on the objects T1–T3,we can choose di?erent weights c ik.If sensor A i is more important to object T k than to T l(k;l21;2;3), then c ik will be larger than c il.The following must be observed when choosing the weights:

?If x ik?0,then c ik?0,to ensure that object T k which is not related to sensor A i is not allocated any sensor resource.

?06c ik61.

?In order that the sensor resources are fully allocated, P3

k?1

c ik?1;i?1;2; (7)

The weight matrix C we decide on for this particular problem is as follows:

3.f ik:

(1)b ijk:

C-GPM has a clear advantage over other approaches in being able to deal with a variety of coordinations occurring in multi-sensor systems.In Section3,we divide typical coordinations into four categories(I,II,III,IV)of12 types.

Since the nose A2is not related with the object T1of opening a door,the coordination of the eye A1with respect to the nose A2is unilateral coordination,whereby A2 would modify its intention,and A1would not.Therefore,

b

121

?II and b211?I.Since the eye A1and the arm A4

Input:c ik;x ik;f ik

Output:

1.Initialization:

t0

r iketT—Initialize in parallel

2.Whileed u ik=d t?0Tdo

t tt1

u iketT—Compute in parallel according to Eq.(3) d u ik=d t—Compute in parallel according to Eq.(8)

d r iketT=d t—Comput

e in parallel according to Eq.

(11)

r iketTr iketà1Ttd r iketT=d t

d c iketT=d t—Comput

e in parallelaccording to Eq.(10) c iketTc iketà1Ttd c iketT=d t x ik T1T2T3

open a door take dictation?nd food and

take back

A1:eye111

A2:nose001

A3:ear010

A4:arm111

A5:hand111

A6:leg101

A7:foot101

c ik T1T2T3

open a door take dictation?nd food and

take back

A1:eye0.50.250.25

A2:nose001

A3:ear010

A4:arm0.40.30.4

A5:hand0.330.340.33

A6:leg0.3500.65

A7:foot0.3500.65

X.Feng et al./Information Fusion9(2008)450–464459

are both related to the object T1of opening a door,but they themselves are unrelated,the coordination between the eye A1and the arm A4is bilateral coordination,and neither A1nor A4would modify their intention.So b141?III and b411?III.Since the arm A4and the hand A5are both related to the object T1of opening a door and they in?uence each other,the coordination between A4and A5is bilateral coordination,and A4and A5must both modify their intention.So b451?IV and

b 541?IV.

As such,we can obtain all the b ijk values,as shown in the following,where for object T k,if there is no coordina-tion between sensor A i and A j,b ijk is set to‘‘–’’.

(2)f ijk:

According to Eq.(6),

f ijketT?

1if b ijk2LeIIT[LeIIITà1if b ijk2LeIT[LeIVT(

f jiketT?

1if b jik2LeIT[LeIIITà1if b jik2LeIIT[LeIVT(

and so we obtain f ijk as follows:

(3)f ik:

According to Eq.(5),f iketT?

P n

j?1

f ijketTt

P n

j?1

f jiketT, we get f ik as follows:

f ik is equal to the sum of the i th row and the i th column

of the matrixef ijkT

7?7

.

4.r ik:

Initialization:et?0T

b ij1A1A2A3A4A5A6A7 A1–II II III III III III A2I––I I I I A3I––I I I I A4III II II–IV III III A5III II II IV–III III A6III II II III III–IV A7III II II III III IV–b ij2A1A2A3A4A5A6A7 A1–II III III III II II A2I–I I I––A3III II–III III II II A4III II III–IV II II A5III II III IV–II II A6I–I I I––A7I–I I I––

b ij3A1A2A3A4A5A6A7 A1–III II III III III III A2III–II III III III III A3I I–I I I I A4III III II–IV III III A5III III II IV–III III A6III III II III III–IV A7III III II III III IV–f ij1A1A2A3A4A5A6A7 A10111111 A2à100à1à1à1à1 A3à100à1à1à1à1 A41110à111 A5111à1011 A6111110à1 A711111à10 f ij2A1A2A3A4A5A6A7 A10111111 A2à10à1à1à100 A31101111 A41110à111 A5111à1011 A6à10à1à1à100 A7à10à1à1à100

f ij3A1A2A3A4A5A6A7 A10111111 A21011111 A3à1à10à1à1à1à1 A41110à111 A5111à1011 A6111110à1 A711111à10 f ik T1T2T3 A18610 A20010 A3060

A4426

A5426

A6406

A7406

r ik T1T2T3 A11/31/31/3 A2001

A3010

A41/31/31/3 A51/31/31/3 A61/201/2 A71/201/2

460X.Feng et al./Information Fusion9(2008)450–464

Here r ik is initialized by the average values.Alterna-tively,r ik can also be initialized as random numbers between0and1.In fact,based on the experiments we have done,we found that the results are not a?ected by the ini-tialization of R.Whatever r ik is initialized to be,the matrix R is dealt with using the following three steps:

?r ik?0j8x ik?0.If x ik?0,then r ik?0,to ensure that an object T k which is not related to the sensor A i is not allo-cated any sensor resource.

?Non-negativity:06r ik61.If min i;k r ik<0,then let

r ik?r ikàmin i;k r ik.?Normalization.Let r ik?r ik=P3

k?1

r ik,in order that the

sensor resources are fully allocated;that is, P3

k?1

r ik?1;i?1;2; (7)

5.zeRT:

According to Eq.(1),zeRT?P n

i?1

P m

k?1

c ik r ik x ik?4:023.

https://www.360docs.net/doc/0410065210.html,pute in parallel(showing only the?rst evo-lutionary iteration;t=1).

1.D r ik:

According to Eq.(11),we have

D r ik%d r ik=d t?k1o u ik

o r ik

tk2

o J

o r ik

àk3

o P

o r ik

àk4

o Q

o r ik

where,

o u ik

o r ik

?c ik x ik expeàc ik r ik x ikTo J

o r ik

?c ik x ik expeàc ik r ik x ikT

o P o r ik ?

o P

o u ik

á

o u ik

o r ik

?àu iká

expeàu2

ik

=2 2T

P n

i?1

P m

k?1

expeàu

ik

=2 2T

á

o u ik

o r ik

o Q o r ik ?2

X m

k?1

x iká

X n

i?1

X m

k?1

r ik x ikà1

1

1texpeàf ik u ikT

à

1

2

!

á

o u ik

o r ik

2.D c ik:

According to Eq.(10),we have

D c ik%d c ik=d t?k1o u ik

o c ik

tk2

o J

o c ik

àk3

o P

o c ik

àk4

o Q

o c ik

where,

o u ik o c ik ?r ik x ik expeàc ik r ik x ikT

o J o c ik ?r ik x ik expeàc ik r ik x ikT

o P o c ik ?

o P

o u ik

á

o u ik

o c ik

?àu iká

expeàu2

ik

=2 2T

P n

i?1

P m

k?1

expeàu2

ik

=2 2T

á

o u ik

o c ik

o Q o c ik ?à

1

1texpeàf ik u ikT

à

1

2

!

á

o u ik

o c ik

3.r iket?1Tand c iket?1T:

r iket?1T?r iket?0TtD r iket?1T

c iket?1T?c iket?0TtD c iket?1T

k1?0:05k2?0:05k3?0:9k4?0:9 ?0:8

After we get r iket?1T,matrix R should be dealt with

using the three steps just mentioned again.

In addition,after we get c iket?1T,matrix C should be

dealt with similarly,i.e.,

?c ik?0j8x ik?0.

?Non-negativity:Let c ik?c ikàmin i;k c ik.

?Normalization.Let c ik?c ik=

P3

k?1

c ik.

Regarding the coe?cients k1,k2,k3,k4and ,we can

draw the following conclusions from the theoretical proofs

and the experiments we have done:

?When is larger,the convergence speed of C-GPM algo-

rithm is faster.

?If the values of k1,k2,k3,and k4change in direct propor-

tion,the experimental results will hardly be in?uenced.

?In order for the C-GPM algorithm to converge,k3and

k4should be much larger than k1and k2,based on

Eqs.(17)and(19).

At the end of the?rst iteration we get

Ret?1T?

0:36920:31110:3197

001:0000

01:00000

0:33680:31580:3474

0:33310:32600:3409

0:458200:5418

0:458200:5418

B B

B B

B B

B B

B B

B B

B B

@

1

C C

C C

C C

C C

C C

C C

C C

A

Cet?1T?

0:48690:24830:2648

001:0000

01:00000

0:36030:26610:3735

0:33040:32690:3427

0:355400:6446

0:355400:6446

B B

B B

B B

B B

B B

B B

B B

@

1

C C

C C

C C

C C

C C

C C

C C

A

zet?1T?4:0690

Obviously,zet?1T>zet?0T,the robot multi-sensor

fusion problem is being optimized.The evolutionary exper-

imental results for z from t=1to t=120are as follows,

and depicted in Fig.5.

As shown in Fig.5,the convergence speed is faster at

the beginning(from t=0to t=20)of the evolution.The

X.Feng et al./Information Fusion9(2008)450–464461

optimization trend of z re?ects exactly the optimization of the problem.At about t =60,z approaches the maximum of 4.53.At t =96,z reaches its maximum,and stays unchanged in the remainder of the iterations.From t =104to 120,R is always equal to

1:00000:00000:0000

01:000001:000000:00000:00001:00000:00000:00001:00000:000001:00000:0000

1:0000

B

B B

B B

B

B B

B B

B

@1C C C C C C C

C C

C C A Since r ik (t =104)=r ik (t =105)and c ik (t =104)=

c ik (t =105),

d r ik /d t (t =105)=0and d c ik /d t (t =105)=0.

Thus,d u ik d t (t =105)=o u ik o r ik (t =105)d r ik d t (t =105)+o u ik

o c ik

(t =105)d c ik

d t (t =105)=0.Therefore,th

e C-GPM algorithm ends appropriately.This shows that sensor resources are allocated to the objects whose weights are the maximum.Undoubtedly this solution o

f R obtained by C-GPM is the optimum solution as it maximizes z ,which veri?es the approach’s convergence and its ability to arrive at the optimum for a complex problem.The solution at t =120however is not a practical solution because in order to achieve the objective,it does not give the ‘‘less important’’sensors any resource.A more practical solution is probably at around t =20where z is very near its maximum value and every sensor is allocated some resource:

0:88520:02770:087100

1:000001:000000:10410:04260:85330:10370:05650:83980:072300:92770:0723

00:9277

B B B B B B B

B B

B B B

B @

1

C C C

C

C

C C

C

C

C C

C

C A It should be possible to re-formulate the problem model,

perhaps with the addition of more constraints,such that the iterative algorithm will settle at an optimum yet practi-cal solution.

Besides the C-GPM algorithm’s good performance in the mathematical sense,the particles’evolution and motion during the process of resolving the problem are also very interesting.If x ik ?1,then s ik is a particle.For our robot problem,there are 15

particles.

Fig.5.Optimization from t =0to t =120.

s ik T 1T 2T 3A 1?

??A 2?A 3?A 4???A 5??

?A 6??A 7

?

?

462X.Feng et al./Information Fusion 9(2008)450–464

According to Eq.(3),u ik ?1àexp eàc ik r ik x ik T,we get u ik et ?0T,as follows:

u ik is the y-coordinate of particle s ik .The x-coordinate of particle s ik represents the ordinal number of the particle.The initial states of the 15particles in the force ?eld are shown in Fig.6.

When t =5,10,15,120,the states of the 15particles in the force ?eld are shown in Figs.7–10,respectively.

5.Conclusion

In this paper,we describe a new evolutionary approach to multi-sensor fusion based on the coordination general-ized particle model (C-GPM).We study some theoretical foundations of the approach including its convergence.The validity of C-GPM is veri?ed via several formal proofs and by simulations.

u ik et ?0TT 1T 2T 3A 10.15210.07920.0792A 200

0.6321A 30

0.63210

A 40.12370.09430.1237A 50.10320.10610.1032A 60.160500.2775A 7

0.1605

0.2775

Fig.6.The initial state of 15particles in the force

?eld.

Fig.7.When t =

5.

Fig.8.When t =

10.

Fig.9.When t =

15.

Fig.10.When t =120.

X.Feng et al./Information Fusion 9(2008)450–464463

We present the C-GPM approach as a new branch of evolutionary algorithms,which can overcome the limita-tions of other existing evolutionary algorithms in capturing the entire dynamics inherent in the problem,especially those that are high-dimensional,highly nonlinear,and random.

Hence,the C-GPM approach can describe the complex behaviors and dynamics of multiple sensors.

The model treats each possible allocation of resources as a con?guration of particles with‘‘physical’’attributes and then?nds the equilibrium of the system iteratively.A novel aspect of the approach is that it takes into account‘‘uncon-scious’’social coordinations between the sensors.

The C-GPM algorithm can work out the theoretical optimum solution,which is important and exciting.To summarize,the C-GPM approach has the following attrac-tive features:

?Very high parallelism and real-time computational performance.

?A sensor in a multi-sensor system is regarded as being neither fully sel?sh nor fully unsel?sh,and so di?erent sensors can exhibit di?erent degrees of autonomy.

?A variety of complicated social coordinations among the sensors can be taken into account in the process of prob-lem-solving.

?The C-GPM algorithm can work out the optimum solu-tion of any multi-sensor fusion problem.

?C-GPM is a physics-based evolutionary algorithm that can overcome the limitations of other traditional evolu-tionary algorithms in describing the high-dimensional, highly nonlinear,and random behaviors and dynamics of objects.

?There are only?ve coe?cients in the C-GPM algorithm, whose values are all easy to choose. Acknowledgements

We thank the editor and referees for their very clear and useful advice and comments.This work is supported by a Hong Kong University Small Project Funding (200607176155).

References

[1]S.Forrest,Genetic algorithms—principles of natural-selection applied

to computation,Science261(5123)(1993)872–878.

[2]S.Kirkpatrick,C.D.Gelatt,M.P.Vecchi,Optimization by simulated

annealing,Science220(4598)(1983)671–680.

[3]E.Bonabeau,M.Dorigo,G.Theraulaz,Inspiration for optimization

from social insect behaviour,Nature406(6791)(2000)39–42.

[4]J.Kennedy,R.C.Eberhart,Particle swarm optimization,in:Proc.

IEEE Conf.Neural Networks,Piscataway,NJ,vol.IV,1995,pp.

1942–1948.

[5]Z.Ren,C.J.Anumba,O.O.Ugwu,The development of a multi-agent

system for construction claims negotiation,Advances in Engineering Software34(2003)683–696.

[6]Z.Ren,C.J.Anumba,Multi-agent systems in construction state of

the art and prospects,Automation in Construction13(2004) 421–434.

[7]G.Vincent,L.P.Christophe,S.Sami,A multi-agents architecture to

enhance end-user individual based modelling,Ecological Modelling 157(2002)23–41.

[8]C.J.Emmanouilides,S.Kasderidis,J.G.Taylor,A random asymmetric

temporal model of multi-agent interactions:dynamical analysis, Physica D181(2003)102–120.

464X.Feng et al./Information Fusion9(2008)450–464

相关主题