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

The coordination generalized particle model—An evolutionaryapproach to multi-sensor fusionXiang Fenga,*,Francis u a ,Dianxun ShuaibaDepartment of Computer Science,The University of Hong Kong,Hong KongbDepartment of Computer Science and Engineering,East China University of Science and Technology,Shanghai 200237,PR ChinaReceived 9November 2006;received in revised form 6January 2007;accepted 6January 2007Available online 14January 2007AbstractThe rising popularity of multi-source,multi-sensor networks supporting real-life applications calls for an efficient 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 difficult 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.IntroductionSensor 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 classified into different 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 different 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 difficult 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],ant1566-2535/$-see front matter Ó2007Elsevier B.V.All rights reserved.doi:10.1016/j.inffus.2007.01.001*Corresponding author.Tel.:+852********;fax:+852********.E-mail address:xfeng@cs.hku.hk (X.Feng)./locate/inffusAvailable online at Information Fusion 9(2008)450–464colony 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 artificial 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 thefield of potential would change the particle’s coordinates and energies.The change in the state of the particle is a result of the influence of the potential.In C-GPM,each particle is described by some differential dynamic equations,and the results of their calculations govern the movement(to a new state in thefield)of the particle.Specifically,each particle computes the combined effect of its own autono-mous self-driving force,thefield 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 differences between our C-GPM algorithm and other popular EA can be seen in Table1.The common features of these different 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 difficult problems.•They consistently perform well infinding approximating solutions to all types of problems.•They are mainly used in thefields of artificial 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 allocationIn a sensor-based application with command and con-trol,a major prerequisite to the success of the command and control process is the effective 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 insufficient resources to perform all the desired tasks,the sensor management must allocate the available sensors to those tasks that could maximize the effectiveness of the sensing process.Table1The C-GPM algorithm vs.other popular EAC-GPM GA SA ACO PSOInspired by Physical models of particledynamics NaturalevolutionThermodynamics Behaviors ofreal antsBiological swarm(e.g.,swarm of bees)Key components Energy function;differentialdynamic equations Chromosomes Energy function Pheromone laid Velocity-coordinatemodelExploration Both macro-evolutionary andmicro-evolutionary processes Macro-evolutionaryprocessesMicro-evolutionaryprocessesMacro-evolutionaryprocessesMacro-evolutionaryDynamics Can capture the entire dynamicsinherent in the problem Cannot capture Can capturepartlyCannot capture Cannot captureHigh-dimensional,highly nonlinear, random behaviors and dynamics Can describe Cannot describe Can describepartlyCannot describe Cannot describe X.Feng et al./Information Fusion9(2008)450–464451The 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 ¼f A 1;...;A n g ,and m objects,T ¼f 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 ðt Þ¼f x 1k ;...;x ik ;...;x nk g ,wherex ik ðt Þ¼1if sensor i is used on object k at interval t0otherwise &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 ¼1r ik ðt Þx ik ðt Þ¼1ð1Þwhere 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 ¼ðc ik Þn Âm be a two-dimensional weight vector.Sensor allocation can be defined as a problem to find a two-dimensional allocation vector R ¼ðr ik Þn Âm ,which max-imizes the objective in (2),subject to the constraint (1)z ðR Þ¼ðC ÞTRX ¼X n i ¼1X m k ¼1c ik r ik x ik ð2ÞLet f ik ðt Þrepresent the intention strength of socialcoordination.Thus we obtain an allocation-related matrix S ðt Þ¼½s ik ðt Þ n Âm ,as shown in Table 2,where s ik ðt Þ¼h r ik ðt Þ;c ik ðt Þ;x ik ðt Þ;f ik ðt Þi :For convenience,both r ik ðt Þand c ik ðt Þare normalized such that 06r ik ðt Þ61and 06c ik ðt Þ61.3.The C-GPM approach to sensor fusion 3.1.Physical model of C-GPMThis 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 ofthe allocation-related matrix S ik as a particle,s ik ,in a force field.The problem solving process is hence transformed into one dealing with the kinematics and dynamics of par-ticles in the force field.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 field.Particle and force-field 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 different from particles in physics.As shown in Fig.1,the vertical coordinate of a particle s ik in force field 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 field,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 field con-tributes an upward component of a particle’s motion,which represents the tendency that the particle pursues the collective benefit 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 reflect 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 profit (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 field,and once they have all reached their respective equilibrium positions,we have a feasible solution to the optimization problem in question.Table 2The matrix S ðt ÞSensors Objects T 1...T k...T mA 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 nr 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 ;fnm452X.Feng et al./Information Fusion 9(2008)450–464Because 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 affect 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-GPMWe define in this subsection the mathematical model of C-GPM for the sensor allocation problem that involves n sensors and m objects.Let u ik ðt Þbe the distance from the current position of particle s ik to the bottom boundary of force field F at time t ,and let J ðt Þbe the utility sum of all particles,which we define as follows:u ik ðt Þ¼a ½1Àexp ðÀc ik ðt Þr ik ðt Þx ik ðt ÞÞJ ðt Þ¼Xn i ¼1X m k ¼1u ik ðt Þð3Þwhere 0<a <1.1Àe Àx is chosen as the definition of u ik be-cause 1Àe Àx is a monotone increasing function and be-tween 0and 1(Fig.2).At time t ,the potential energy functions,P ðt Þ,which is caused by the upward gravitational force of force field F ,is defined byP ðt Þ¼ 2ln X n i ¼1X m k ¼1exp ½Àu 2ik ðt Þ=2 2 À 2ln mnð4Þwhere 0< <1.The smaller P ðt Þis,the better.With Eq.(4),we attempt to construct a potential energy function,P ðt Þ,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 field.The bigger is,the better.If we did not get a sufficiently satisfactory result by C-GPM,we can let smaller.The gravitational force of the force field 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 beneficial consequence possibly caused by A j ,A i wants to change its own current intention.C ijk :To benefit A j ,A i wants to change its own current intention regardless of self-benefit.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 beneficial 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 benefit 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 ;forFig.2.Graphical presentation of u ik ðt Þ.Table 3Social coordinations among sensors b ijk Type Namef ijk IA ijk Adaptive avoidance coordination À1B ijk Adaptive exploitation coordinationC ijk Collaboration coordination IID ijk Tempting avoidance coordination 1E ijk Tempting exploitation coordinationF ijk Deception coordinationIII G ijk Competition coordination 1H ijk Coalition coordinationI ijk Habituation/preference coordination IV J ijk Antagonism coordination À1K ijk Reciprocation coordination L ijkCompromise coordinationX.Feng et al./Information Fusion 9(2008)450–464453G 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 haveLðIÞ¼Lð10Þ¼f A ijk;B ijk;C ijk j8i;j;k g;LðIIÞ¼Lð01Þ¼f D ijk;E ijk;F ijk j8i;j;k g;LðIIIÞ¼Lð00Þ¼f G ijk;H ijk;I ijk j8i;j;k g;LðIVÞ¼Lð11Þ¼f J ijk;K ijk;L ijk j8i;j;k g;L¼ðLð01Þ[Lð10Þ[Lð10Þ[Lð11ÞÞ:The intention strength f ikðtÞof sensor A i with respect to object T k is defined byf ikðtÞ¼X nj¼1f ijkðtÞþX nj¼1f jikðtÞð5Þf ijkðtÞ¼1if b ijk2LðIIÞ[LðIIIÞÀ1if b ijk2LðIÞ[LðIVÞ(f jikðtÞ¼1if b jik2LðIÞ[LðIIIÞÀ1if b jik2LðIIÞ[LðIVÞ(ð6Þb 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 ijkðtÞof intention strength f ikðtÞ.f ikðtÞof s ikðtÞrepresents the aggregate intention strength when more than one social coordination happen simulta-neously at time t.The greater f ikðtÞis,the more necessary would sensor A i have to modify its r ikðtÞfor object T k.At time t,the potential energy function,QðtÞ,is defined byQðtÞ¼X ni¼1X mk¼1r ikðtÞx ikðtÞÀ12ÀXi;k Z u ikf½1þexpðÀf ik xÞ À1À0:5g d xð7ÞThefirst term of QðtÞis 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).Thefirst term of QðtÞcorresponds 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 QðtÞis chosen as shown because we want o Qikto be a monotone decreasing sigmoid function,as shown in Fig.3.Àf½1þexpðÀf ik u ikÞ À1À0:5g is such a function.Thereforewe let o Qo u ik equal toÀf½1þexpðÀf ik u ikÞ À1À0:5g.Then o Qo u ikisintegrated to be Q.A particle in the forcefield can move upward along a vertical line under a composite force making up of•the upward gravitational force of the forcefield,•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 forcefield.The general hybrid potential energy function for particle s ik, E ikðtÞ,can be defined byE ikðtÞ¼k1u ikðtÞþk2JðtÞÀk3PðtÞÀk4QðtÞð8Þwhere0<k1;k2;k3;k4<1:Dynamic equations for particle s ik is defined byd u ikðtÞ=d t¼W1ðtÞþW2ðtÞW1ðtÞ¼Àu ikðtÞþc v ikðtÞW2ðtÞ¼k1þk2o JðtÞikÀk3o PðtÞikÀk4o QðtÞikh io u ikðtÞikh i2þo u ikðtÞikh i2&'8>>>>>><>>>>>>:ð9Þwhere c>1.And v ikðtÞis a piecewise linear function of u ikðtÞdefined byv ikðtÞ¼0if u ikðtÞ<0u ikðtÞif06u ikðtÞ611if u ikðtÞ>18><>:ð10ÞIn order to dynamically optimize sensor allocation,the particle s ik may alternately modify r ik and c ik,respectively, as follows:d c ikðtÞ=d t¼k1o u ikðtÞo c ikðtÞþk2o JðtÞo c ikðtÞÀk3o PðtÞo c ikðtÞÀk4o QðtÞo c ikðtÞð11Þd r ikðtÞ=d t¼k1o u ikðtÞo r ikðtÞþk2o JðtÞo r ikðtÞÀk3o PðtÞo r ikðtÞÀk4o QðtÞo r ikðtÞð12Þwhere o QðtÞik¼Àf½1þexpðÀf ikðtÞu ikðtÞÞ À1À0:5g is a sig-moid function of the aggregate intention strength f ikðtÞ. The graphical presentation of o Qikis shown in Fig.3.Note that f ikðtÞis related to social coordinations amongthe sensors at time t.o Qo u ikis a monotone decreasing function. By Eq.(9)the greater the value of f ikðtÞ,the more thereare Fig.3.Graphical presentation of o Qo u ik.454X.Feng et al./Information Fusion9(2008)450–464values of W 2ðt Þand hence u ik ðt Þwill increase,which implies that the social coordinations will strengthen the current allocation r ik ðt Þ.Because o Q o u ikis a monotone decreasing function,the greater the value of f ik ðt Þ,the smaller o Q ik ,the greater Ào Q ik,the greater W 2ðt Þby Eq.(9)and the greater D r ik ðt þ1Þby Eq.(12).Since r ik ðt þ1Þ¼r ik ðt ÞþD r ik ðt þ1Þ,the greater the value of f ik ðt Þ,the greater r ik ðt þ1Þ.Since u ik is a monotone increasing function,the greater r ik ðt þ1Þ,the greater u ik f ik ðt Þ")o Q o u ik #)Ào Q o u ik ")ð9ÞW 2"Ào Q o u ik")ð11ÞD r ik ðt þ1Þ")ar ik ðt þ1Þ")ð3Þbu ik "(a)r ik ðt þ1Þ¼r ik ðt ÞþD r ik ðt þ1Þ;(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 2ðt Þof Eq.(9).Denote the j th terms of Eqs.(11)and (12)by d c ik ðt ÞD E j and d r ik ðt ÞD E j;respectively.When allotted resource r ik is updated according to (12),the first and second terms of (12)will cause the following speed increments of the parti-cle s ik ,respectively:h d u ik ðt Þ=d t i r1¼o u ik ðt Þo r ik ðt Þd r ik ðt Þd t ()1¼k 1o u ik ðt Þo r ik ðt Þ !2ð13Þh d u ik ðt Þ=d t i r2¼o u ik ðt Þo r ik ðt Þd r ik ðt Þd t ()2¼k 2o u ik ðt Þo r ik ðt Þo J ðt Þo r ik ðt Þ¼k 2o u ik ðt Þo r ik ðt Þo J ðt Þo u ik ðt Þo u ik ðt Þo r ik ðt Þ¼k 2o J ðt Þo u ik ðt Þo u ik ðt Þo r ik ðt Þ !2ð14ÞSimilarly,the third and the fourth term of Eq.(12)will cause the following speed increments of the particle s ik :h d u ik ðt Þ=d t i r 3¼Àk 3o P ðt Þik o u ik ðt Þik !2h d u ik ðt Þ=d t i r4¼Àk 4o Q ðt Þo u ik ðt Þo u ik ðt Þo r ik ðt Þ!2Similarly,for Eq.(11),we have h d u ik ðt Þ=d t i cj ;j ¼1;2;3;4.We thus obtainX 4j ¼1½h d u ik ðt Þ=d t i c j þh d u ik ðt Þ=d t i r j¼k 1þk 2o J ðt Þo u ik ðt ÞÀk 3o P ðt Þo u ik ðt ÞÀk 4o Q ðt Þo u ik ðt Þ!o u ik ðt Þo r ik ðt Þ!2(þo u ik ðt Þo c ik ðt Þ!2)¼W 2ðt Þ:Therefore,updating c ðj Þik and r ðj Þik by (11)and (12),respec-tively,gives rise to the speed increment of particle s ik that is exactly equal to W 2ðt Þof Eq.(9).Proposition 2.The first 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 ðk 1þk 2Þ.According to Eqs.(13)and (14),the sum of the first and second terms of Eqs.(11)and (12)will beh d u ik ðt Þ=d t i r 1þh d u ik ðt Þ=d t i r 2þh d u ik ðt Þ=d t i c 1þh d u ik ðt Þ=d t i c2¼k 1þk 2o J ðt Þo u ik ðt Þ !o u ik ðt Þo r ik ðt Þ !2þo u ik ðt Þo c ik ðt Þ !2()¼ðk 1þk 2Þx 2ik ðt Þ½r 2ik ðt Þþc 2ik ðt Þ ½Àu ik ðt Þ 2P 0:Therefore,the first and second terms of (11)and (12)willcause u ik ðt Þto monotonically increase.Proposition 3.For C-GPM,if is very small,then decreas-ing the potential energy P ðt Þof Eq.(4)amounts to increasing the minimal utility of an sensor with respect to an object,min-imized over S ðt Þ.Supposing that H ðt Þ¼max i ;k fÀu 2ik ðt Þg ,we have½exp ðH ðt Þ=2 2Þ 2 26X n i ¼1X m k ¼1exp ðÀu 2ik ðt Þ=2 2Þ"#2 26½mn exp ðH ðt Þ=2 2Þ 2 2:Taking the logarithm of both sides of the above inequalities givesH ðt Þ62 2ln X n i ¼1X m k ¼1exp ðÀu 2ik ðt Þ=2 2Þ6H ðt Þþ2 2ln mn :Since mn is constant and is very small,we haveH ðt Þ%2 2ln X n i ¼1X m k ¼1exp ðÀu 2ik ðt Þ=2 2ÞÀ2 2ln mn ¼2P ðt Þ:It turns out that the potential energy P ðt Þat the time t rep-resents the maximum of Àu 2ik ðt Þamong 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 ðt Þwill result in the increase of the minimum of u ik ðt Þ.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 ðt Þ,is given by d u ik ðt Þd t ()3¼h d u ik ðt Þ=d t i r 3þh d u ik ðt Þ=d t i c3¼Àk 3o P ðt Þo u ik ðt Þo u ik ðt Þo r ik ðt Þ !2þo u ik ðt Þo c ik ðt Þ!2():X.Feng et al./Information Fusion 9(2008)450–464455。