Research ArticleNovel Particle Swarm Optimization and Its Application inCalibrating the Underwater Transponder Coordinates

Zheping Yan Chao Deng Benyin Li and Jiajia Zhou

College of Automation Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Chao Deng soledc163com

Received 25 November 2013 Accepted 3 March 2014 Published 17 April 2014

Academic Editor P Karthigaikumar

Copyright copy 2014 Zheping Yan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A novel improved particle swarm algorithm named competition particle swarm optimization (CPSO) is proposed to calibrate theUnderwater Transponder coordinates To improve the performance of the algorithm TVAC algorithm is introduced into CPSOto present an extension competition particle swarm optimization (ECPSO) The proposed method is tested with a set of 10 standardoptimization benchmark problems and the results are comparedwith those obtained through existing PSO algorithms basic particleswarm optimization (BPSO) linear decreasing inertia weight particle swarm optimization (LWPSO) exponential inertia weightparticle swarm optimization (EPSO) and time-varying acceleration coefficient (TVAC) The results demonstrate that CPSO andECPSO manifest faster searching speed accuracy and stability The searching performance for multimodulus function of ECPSOis superior to CPSO At last calibration of the underwater transponder coordinates is present using particle swarm algorithm andnovel improved particle swarm algorithm shows better performance than other algorithms

1 Introduction

Particle swarm optimization (PSO) technique is consideredas one of the modern heuristic algorithms for optimizationfirst proposed by Kennedy and Eberhart in 1995 [1] Themotivation for the development of this method was basedon the simulation of simplified animal social behaviors [2]The PSO algorithm works on the social behavior of particlesin the swarm In PSO the population dynamics simulatesa bird flockrsquos behavior where social sharing of informationtakes place and individuals can profit from the discoveriesand previous experience of all other companions during thesearch for food That is the global best solution is found bysimply adjusting the trajectory of each individual towardsits own best location and towards the best particle of theentire swarm at each time step [1ndash3] Owing to its reductionon memory requirement and computational efficiency withconvenient implementation it has gained lots of attentionin various optimal control system applications compared toother evolutionary algorithms [4] Several researches werecarried out so far to analyze the performance of the PSOwith different settings for example Shi and Eberhart [5]indicated that the optimal solution can be improved by

varying the value of 120596 from 09 at the beginning of the searchto 04 at the end of the search for most problems and theyintroduced a method named TVIW with a linearly varyinginertia weight over the generations Chen et al [6] introducedexponential inertia weight strategies which is found to bevery effective for TVIW Ratnaweera et al [2] propose time-varying acceleration coefficients as a parameter automationstrategy for the PSOnamedTVACwitch reduce the cognitivecomponent and increase the social component by changingthe acceleration coefficients with time Ni and Deng [7]analyze the performance of PSO with the proposed randomtopologies and explore the relationship between populationtopology and the performance of PSO from the perspectiveof graph theory characteristics in population topologiesNoel [8] presents a new hybrid optimization algorithm thatcombines the PSO algorithm and gradient-based local searchalgorithms to achieve faster convergence and better accuracyof final solution without getting trapped in local minimaEpitropakis et al [9] motivated by the behavior and spatialcharacteristics of the social and cognitive experience of eachparticle in the swarm develop a hybrid framework that com-bines the particle swarm optimization and the differential

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 672412 12 pageshttpdxdoiorg1011552014672412

2 Mathematical Problems in Engineering

evolution algorithm In an attempt to efficiently guide theevolution and enhance the convergence the author evolvedthe personal experience or memory of the particles withthe differential evolution algorithm without destroying thesearch capabilities of the algorithmMousa et al [10] proposea hybrid multiobjective evolutionary algorithm combininggenetic algorithm and particle swarm optimization the localsearch scheme is implemented as a neighborhood searchengine to improve the solution quality where it intendsto explore the less-crowded area in the current archive topossibly obtain more nondominated solutions

As a kind of optimization algorithm PSO is simple instructure has good performance and is easy to implement Itis widely applied in various engineering applications Moradiand Abedini [11] combined genetic algorithm and particleswarm optimization for optimal location and sizing of dis-tributed generation on distribution systemsThe algorithm isto minimize network power losses make better voltage regu-lation and improve the voltage stabilitywithin the frameworkof system operation and security constraints in radial distri-bution systems Chang et al [12] apply the PSO algorithmto estimate the parameters of the Genesio-Tesi nonlinearchaotic systems and the estimation of the PSO algorithmis verified by examining different sets of random initialpopulations under the presence of measurement noise Soonand Low [13] proposed a new approach using particle swarmoptimization with inverse barrier constraint to determinethe unknown photovoltaic model parameters The proposedmethod has been validated with three different photovoltaictechnologies Jiang et al [14] proposed the barebone particleswarm optimization algorithm to determine the parametersof solid oxide fuel cell (SOPC) The cooperative coevolutionstrategy is applied to divide the output voltage functioninto four subfunctions based on the interdependence amongvariables To the nonlinear characteristic of SOPC model ahybrid learning strategy is proposed for BPSO to ensure agood balance between exploration and exploitation Alfi [15]proposed novel particle swarm optimization to cope with theonline system parameter identification problem The inertiaweight for every particle is dynamically updated based on thefeedback taken from the fitness of the best previous positionfound by the particle and a novel methodology is incorpo-rated into the novel particle swarm optimization to be ableto effectively response and detect any parameter variationsof system to be identified Hu and Shi [16] to solve the pre-mature convergence problem of PSO improved algorithmswith hybrid and mutation operators leading to obtaininga high level of particle population diversity decreasing thepossibility of falling into local optima and improving locationaccuracy The novel algorithm is introduced in the range-based location for wireless sensor networks and simulationshows a better performance than basic PSO algorithm

With the development of marine economy and technol-ogy unmanned underwater vehicle (UUV) is an effectivemeans for marine detection resource exploitation militaryinterfere and investigation [17ndash19] Navigation of UUVhas been and remains a substantial challenge to platformsOne of the main driving factors is the ability to carryout long-duration missions fully autonomously and without

supervision from a surface ship [20 21] Combined withinertial navigation the use of one or several transponderson the seabed is an accurate and cost-effective approachtoward solving several of these challenges [22ndash24] It isobvious that the exact position of the transponder is veryimportant in the underwater transponder positioning system[25 26] However in the practical operations due to theinfluence of ocean currents and other factors the practicalcoordinates of transponder will drift from the position whereit launched into the water So it requires the mother shipto calibrate the coordinate of the transponder this paperproposed the particle swarm optimization algorithm solvingthe transponder coordinates

The contribution of this paper is concluded as the fol-lowing Firstly considering the competition particle swarmalgorithm each particle will evolve along two differentdirections to generate two homologous particlesThe optimalone is kept through comparing the cost functions of twohomologous particles and the next generation particle willbe obtained finally Secondly according to the advantage ofTVAC combining CPSO and TVAC ECPSO algorithm ispresentedWith a large cognitive component and small socialcomponent at the beginning on the other hand a smallcognitive component and a large social component allow theparticles to converge to the global optima in the latter partof the optimization Simultaneously the evolution for eachparticle at any time is along two different inertia directionsto generate two homologous particles and to obtain nextgeneration particle Lastly ECPSO is introduced to calibratethe coordinate of the transponder

The rest of this paper is organized as follows In Section 2the basic PSOand its previous developments are summarizedIn Section 3 the competition particle swarm optimizationalgorithm and extension competition particle swarm opti-mization algorithm are introduced The experimental set-tings for the benchmark functions and simulation strategiesare explained and the conclusion is drawn based on thecomparison analysis In Section 4 ECPSO is introduced tocalibrate the coordinates of the transponder and simulationsare designed to verify the feasibility of the algorithm present

2 Some Previous Work

Introduced by Dr Kennedy and Dr Eberhart in 1995 PSOhas ever since turned out to be a competitor in the field ofnumerical optimization and there has been a considerableamount of work done in developing the original version ofPSO In this section we summarize some entire significantprevious developments

21 Basic Particle Swarm Optimization (BPSO) In PSO eachsolution called a ldquoparticlerdquo flies in the search space searchingfor the optimal position to land PSO system combineslocal search method (through individual experience) withglobal search methods (through neighboring experience)attempting to balance exploration and exploitation [27] Eachparticle has a position vector 119909

119894(119896) a velocity vector V

119894(119896) the

position with the best fitness encountered by the particle and

Mathematical Problems in Engineering 3

the index of the best particle in the swarmTheposition vectorand the velocity vector of the 119894th particle in the119889-dimensionalsearch space can be represented as 119909

119894= (1199091198941 1199091198942 1199091198943 119909

119894119889)

and V119894= (V1198941 V1198942 V1198943 V

119894119889) respectively The best position

of each particle (119901best) is 119901119894= (1199011198941 1199011198942 1199011198943 119901

119894119889) and

the fitness particle found so far at generation 119896 (119892best) is119901119892= (1199011198921 1199011198922 119901

119892119889) In each generation each particle is

updated by the following two equations

V119894119889 (119896 + 1) = V

119894119889 (119896) + 1198881

times 1199031times (119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 1198882times 1199032times (119901119892119889 (

119896) minus 119909119894119889 (119896))

(1)

119909119894119889 (119896 + 1) = 119909119894119889 (

119896) + V119894119889 (119896 + 1) (2)

The parameters 1198881and 1198882are constants known as acceler-

ation coefficients 1199031and 1199032are random values in the range

from 0 to 1 and the value of 1199031and 1199032is not the same for

every iteration Kennedy and Eberhart [1] suggested settingeither of the acceleration coefficients at 2 in order tomake themean of both stochastic factors in (1) unity so that particleswould over fly only half the time of search The first equationshows that in PSO the search toward the optimum solutionis guided by the previous velocity the cognitive componentand the social component

Since the introduction of the particle swarm optimiza-tion numerous variations of the algorithm have been devel-oped in the literature Eberhart and Shi showed that PSOsearches for wide areas effectively but tends to lack localsearch precision They proposed in that work a solution byintroducing 120596 an inertia factor In this paper we name it asbasic particle swarm optimization (BPSO)

V119894119889 (119896 + 1) = 120596 times V

119894119889 (119896) + 1198881

times 1199031times (119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 1198882times 1199032times (119901119892119889 (

119896) minus 119909119894119889 (119896))

119909119894119889 (119896 + 1) = 119909119894119889 (

119896) + V119894119889 (119896 + 1)

(3)

22 Time-Varying Inertia Weight (TVIW) The role of theinertia weight 120596 is considered very important in PSO conver-gence behavior The inertia weight is applied to control theimpact of the previous history of velocities on the currentvelocity large inertia weight facilitates global explorationwhile small one tends to facilitate local exploration In orderto assure that the particles converge to the best point inthe course of the search Shi and Eberhart [28] have foundthat time-varying inertia weight (TVIW) has a significantimprovement in the performance of PSO and proposedlinear decreasing inertia weight PSO (LWPSO) with a lineardecreasing value of 120596 This modification can increase theexploration of the parameter space during the initial searchiterations and increase the exploitation of the parameter spaceduring the final steps of the search [29] The mathematicalrepresentation of inertia weight is given as follows

120596 = (1205961minus 1205962) times (

MAXITER minus 119896MAXITER

) + 1205962 (4)

where 1205961and 120596

2are the initial and final values of the inertia

weight respectively 119896 is the current iteration number and

MAXITER is the maximum number of allowable iterationsShi and Eberhart [5] indicate that the optimal solution can beimproved by varying the value of 120596 from 09 at the beginningof the search to 04 at the end of the search formost problems

Chen et al [6] proposed natural exponential (base 119890)inertia weight strategies named EPSO and expressed as

120596 = 1205962+ (1205961minus 1205962) times exp[minus( 119870

(MAXITER4))

2

] (5)

23 Time-Varying Acceleration Coefficient (TVAC) In PSOthe particle was updated due to the cognitive component andthe social component Therefore proper control of these twocomponents is very important to find the optimum solutionaccurately and efficiently Ratnaweera et al [2] introduced atime-varying acceleration coefficient (TVAC) which reducesthe cognitive component and increases the social componentby changing the acceleration coefficients 119888

1and 119888

2with

the time evolution The objective of this development is toenhance the global search in the early part of the optimizationand to encourage the particles to converge toward the globaloptima at the end of the search The TVAC is representedusing the following equations

1198881= (119888max minus 119888min)

119896

MAXITR+ 119888min

1198882= (119888min minus 119888max)

119896

MAXITR+ 119888max

(6)

where 119888min and 119888max are constants 119896 is the current iterationnumber and MAXITR is the maximum number of allowableiterations

Simulations were carried out with numerical bench-marks to find out the best ranges of values for 119888

1and 1198882

From the results it was observed that the best solutions weredetermined when changing 119888

1from 25 to 05 and changing 119888

2

from 05 to 25 over the full search range

3 Proposed New Developments

It is clarified from (1) that particlersquos new velocity is correlatedwith three terms the particlersquos previous velocity the valueof the cognitive component and the value of the socialcomponent Therefore proper control method with inertiaweight factor and acceleration coefficients is significant tofind the optimum solution accurately and efficiently

The inertia weight is utilized to adjust the influence of theprevious velocity on the current velocity and balance betweenglobal and local exploration abilities of the ldquoflying particlerdquo[30 31] A larger inertia weight implies stronger globalexploration ability advocating the particle to escape from alocal minimum A smaller inertia weight leads to strongerlocal exploration ability confining the particle searchingwithin a local range near its present position to guarantee theconvergence

Kennedy and Eberhart [1] indicated that a relatively highvalue of the cognitive component compared with the socialcomponent will result in excessive wandering of individuals

4 Mathematical Problems in Engineering

through the search space In contrast a relatively high value ofthe social component may lead particles to rush prematurelytoward a local optimum

Considering those concerns we propose a new strategyfor the PSO concept

31 Competition Particle Swarm Optimization (CPSO) Inthe process of particle evolution each particle is evolvedalong different directions with different inertia coefficientsand acceleration coefficients Two homologous particles aregenerated and the optimal one is kept through comparingcost functions of two homologous particles eliminating theinferior one Then the next generation particle is updatedfinally The evaluation function of each particle is describedas

V119905119894119889(119896 + 1) = 120596

119905V119894119889 (119896) + 119888

119905

1119903119905

1(119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 119888119905

2119903119905

2(119901119892119889 (

119896) minus 119909119894119889 (119896))

119909119905

119894119889(119896 + 1) = 119909

119905

119894119889(119896) + V119905

119894119889(119896 + 1)

(7)

And the final equations are shown as

V119894119889 (119896) = V119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

119909119894119889 (119896) = 119909

119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

(8)

where 119899 is the number of particles in the swarm 119872 is themaximum iteration frequency 119903119905

1 1199031199052are the randomnumbers

in the range of [0 1] 119909119894119889(119896) is the 119894th particle position in the

119889th dimension after 119896 time iteration 119909119905119894119889(119896) shows the 119889th

dimension position for subparticle 119905 of the 119894th particle after119896 time iteration 120596119905 is the speed inertia weight of subparticle119905 1198881199051and 1198881199052are constants denoting acceleration coefficients

fitness (119909119905119894(119870)) is the fitting function of the subparticle 119905 after

119896 time iteration V119905119894119889(119896) is the speed of subparticle 119905 at 119889th

dimension 119901119894119889(119896) is the optimal position of the 119894th particle

at 119889th dimension and 119901119892119889(119896) is the swarm optimal position

at 119889th dimension after 119896 time iteration

Remark 1 In this paper two subparticles are generated foreach particle at one time therefore 119905 = 1 2

The detailed steps are shown as follows

Step 1 Initial

Substep 1 Set the initial parameters 119899119872 119908119905 1198881199051 1198881199052

Substep 2 Take random initial 119909119894119889(0)

Substep 3 Take random initial V119905119894119889(1)

Substep 4 Calculate fitness(119909119905119894(0)) and set 119901

119894(0) = 119909

119894(0)

Substep 5 One has 119901119892(0) = 119909

119894(0) | 119909

119894(0) isin

min(fitness(119909119894(0)))

Step 2 If the criteria are satisfied output the best solutionotherwise go to Substep 6

Substep 6 Update V119905119894119889(119896) and 119909119905

119894119889(119896)

Substep 7 Calculate fitness(119909119905119894(119896))

Substep 8 One has 119909119894119889(119896) = 119909

119905

119894119889(119896) | min(fitness(119909119905

119894(119896))) 119905 =

1 2

Substep 9 If fitness(119909119894(119896)) lt fitness(119901

119894(119896))

119901119894 (119896) = 119909119894 (

119896) (9)

If minfitness(119909119894(119896)) 119894 = 1 119899 lt fitness(119901

119892(119896))

119901119892 (119896) = 119909119894 (

119896) | min [fitness (119909119894 (119896)) 119894 = 1 119899] (10)

Substep 10 Go back to Step 2

32 Extension Competition Particle Swarm Optimization(ECPSO) competition particle swarm optimization (CPSO)helps adjust the search direction particles and improve thesearch speed and efficiency but due to rapid convergenceCPSO is easy to fall into local minima According tobenchmark functions simulation in Section 4 it is obviousthat CPSO is superior to the searching effective of single-modulus function and TVAC is superior to the searchingeffective of multimodulus function The reason resulting inthis phenomenon is due to the selection of accelerationcoefficient With a large cognitive component and a smallsocial component at the beginning particles are allowed tomove around the search space instead of moving towardthe population best On the other hand a small cognitivecomponent and a large social component allow the particlesto converge to the global optima in the latter part of theoptimization Considering the advantage of TVAC introduceTVAC into CPSO and the extension competition particleswarm optimization (ECPSO) with the acceleration coeffi-cients proposed as above The evolution equations can bemathematically represented as the following

V119905119894119889(119896 + 1) = 120596

119905V119894119889 (119896) + 1198881

119903119905

1(119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 1198882119903119905

2(119901119892119889 (

119896) minus 119909119894119889 (119896))

119909119905

119894119889(119896 + 1) = 119909

119905

119894119889(119896) + V119905

119894119889(119896 + 1)

(11)

where

1198881= (119888max minus 119888min)

119896

MAXITR+ 119888min

1198882= (119888min minus 119888max)

119896

MAXITR+ 119888max

(12)

And it is obvious that

V119894119889 (119896) = V119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

119909119894119889 (119896) = 119909

119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

(13)

Mathematical Problems in Engineering 5

4 Experimental Settings and SimulationStrategies for Benchmark Testing

Simulations were carried out to observe the rate of con-vergence and the quality of the optimum solution of thenew methods introduced in this investigation by comparingwith BPSO EPSO and TVAC From the standard set ofbenchmark problems available in the literature there are 5important functions considered to test the efficacy of theproposed method All of the benchmark functions reflectdifferent degrees of complexity

41 Functions Introduction The functions are as follows

(1) Sphere function one has

1198911 (119909) =

119863

sum

119894=1

1199092

119894 (14)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is verysimple convex and unimodal function with only one localoptimum value

(2) Axis parallel hyperellipsoid function one has

1198912 (119909) =

119863

sum

119894=1

119894 sdot 1199092

119894 (15)

It is known as the weighted sphere model With the searchspace 119909

119894| minus100 lt 119909

119894lt 100 the global minimum locates at

119909 = [0 0]119863 with 119891(119909) = 0 It is continuous convex and

unimodal

(3) Rotated hyperellipsoid function (Schwefelrsquos problem12) one has

1198913 (119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

(16)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is

continuous convex and unimodal With respect to the coor-dinate axes this function produces rotated hyperellipsoids

(4) Moved axis parallel hyperellipsoid function one has

1198914 (119909) =

119863

sum

119894=1

5119894 sdot 1199092

119894 (17)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909(119894) = 5 lowast 119894 119894 = 1 119863 with 119891(119909) = 0

(5) Rosenbrock function one has

1198915 (119909) =

119863minus1

sum

119894=1

[100(119909119894+1minus 1199092

119894)

2

+ (119909119894minus 1)2] (18)

Table 1 Parameters for simulation

Method Parameters

CPSO1205961= 120596max = 09 120596

2= 120596min = 04

1198881

1= 05 1198881

2= 25 1198882

1= 25

1198882

2= 05

119899 = 30119863 = 10

119909119894isin [minus100 100]

V119894isin [minus100 100]

ECPSO1205961= 120596max = 09 120596

2= 120596min = 04

119888min = 05 119888max = 25

BPSO 1198881= 1198882= 20 120596 = 07

EPSO1198881= 1198882= 20 120596max = 09120596min = 04

TVAC 119888min = 05 119888max = 25 120596 = 07

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [1 1]119863 with 119891(119909) = 0 It

is a unimodal function and the global optimum is inside along narrow parabolic shaped flat valley To find the valley istrivial

(6) Rastrigin function one has

1198916 (119909) =

119863

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894) + 10] (19)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is

highly multimodal However the locations of the minima areregularly distributed

(7) Griewank function one has

1198917 (119909) =

1

4000

119863

sum

119894=1

1199092

119894minus

119863

prod

119894=1

cos(119909119894

radic119894

) + 1 (20)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It

is a multimodal function and has many widespread localminima However the locations of the minima are regularlydistributed

(8) Sum of different power function one has

1198918 (119909) =

119863

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816

(119894+1) (21)

The sum of different powers is a commonly used uni-modal test function With the search space 119909

119894| minus100 lt

119909119894lt 100 the global minimum locates at 119909 = [0 0]

119863

with 119891(119909) = 0

(9) Ackleyrsquos path function one has

1198919 (119909) = minus119886 sdot 119890

minus119887sdotradicsum119863

119894=11199092

119894119863minus 119890sum119863

119894=1cos(119888sdot119909119894)119863

+ 119886 + 1198901

(22)

6 Mathematical Problems in Engineering

Table 2 Comparison of BPSO WPSO EPSO TVAC and CPSO

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198911

10 47979119890 minus 11

(11343119890 minus 10)32219119890 minus 24

(10706119890 minus 23)87747119890 minus 51

(3408119890 minus 50)14826119890 minus 54

(14770119890 minus 53)41023119890 minus 146

(36388119890 minus 145)41221119890 minus 118

(30916119890 minus 117)

30 53768(43459)

40137119890 minus 05

(5926119890 minus 05)23219119890 minus 12

(11495119890 minus 11)11062119890 minus 10

(73302119890 minus 10)14872119890 minus 46

(64922119890 minus 46)16829119890 minus 10

(10243119890 minus 09)

50 28412119890 + 01

(45047)02149(01634)

22215119890 minus 05

(35696119890 minus 05)00019(00079)

93999119890 minus 20

(40558119890 minus 19)00022(00083)

70 44686119890 + 01

(43454)11728119890 + 01

(12113119890 + 01)00192(00238)

00609(01033)

20133119890 minus 07

(19424119890 minus 06)00302(00458)

1198912

10 37589119890 minus 10

(17479119890 minus 09)85629119890 minus 24

(19031119890 minus 23)52910119890 minus 50

(23037119890 minus 49)15947119890 minus 52

(14618119890 minus 51)89632119890 minus 146

(35734119890 minus 145)30640119890 minus 113

(30622119890 minus 112)

30 378269(313869)

00005(00007)

10278119890 minus 11

(20979119890 minus 11)11937119890 minus 09

(65345119890 minus 09)81119119890 minus 45

(44882119890 minus 44)30047119890 minus 10

(21285119890 minus 09)

50 68495119890 + 02

(10535119890 + 02)28041(22863)

00004(00010)

01214(06006)

24666119890 minus 15

(24614119890 minus 14)00223(00835)

70 14952119890 + 03

(18073119890 + 02)12961119890 + 02

(14505119890 + 02)02601(02405)

21146(39189)

12547119890 minus 05

(00001)11879(20240)

1198913

10 00013(00018)

40624119890 minus 08

(11066119890 minus 07)31542119890 minus 15

(19686119890 minus 14)36072119890 minus 27

(16376119890 minus 26)60846119890 minus 60

(58552119890 minus 59)44092119890 minus 51

(31687119890 minus 50)

30 20647119890 + 01

(79831)37596(20255)

05809(03373)

00091(00197)

27957119890 minus 06

(50455119890 minus 06)00258(00385)

50 72983119890 + 01

(28263119890 + 01)26189119890 + 01

(13125119890 + 01)10295119890 + 01

(51265)04883(03021)

00775(00620)

05720(03509)

70 14821119890 + 02

(46282119890 + 01)69337119890 + 01

(33718119890 + 01)29336119890 + 01

(10417119890 + 01)27039(10853)

11214(04917)

22342(07423)

1198914

10 27409119890 minus 09

(15090119890 minus 08)13708119890 minus 22

(65937119890 minus 22)35012119890 minus 49

(28064119890 minus 48)99156119890 minus 52

(69209119890 minus 51)16140119890 minus 144

(14296119890 minus 143)30561119890 minus 115

(21383119890 minus 114)

30 19825119890 + 02

(17202119890 + 02)00025(00029)

73319119890 minus 11

(23165119890 minus 10)66962119890 minus 09

(36744119890 minus 08)10828119890 minus 42

(89052119890 minus 42)56960119890 minus 08

(39998119890 minus 07)

50 32922119890 + 03

(56799119890 + 02)13397119890 + 01

(94756)00015(00019)

03064(12614)

22477119890 minus 17

(14523119890 minus 16)02379(11928)

70 74756119890 + 03

(75131119890 + 02)68348119890 + 02

(61444119890 + 02)16799(29749)

10304119890 + 01

(14839119890 + 01)19407119890 minus 06

(96092119890 minus 06)55072

(10286119890 + 01)

1198915

10 56418(1092)

43037(12401)

31701(12637)

07262(09490)

06503(14786)

06445(14706)

30 12322119890 + 03

(94694119890 + 02)42313119890 + 01

(26596119890 + 01)31052119890 + 01

(17699119890 + 01)23370119890 + 01

(19238)14702119890 + 01

(31506)19234119890 + 01

(29573)

50 65575119890 + 03

(13919119890 + 03)29055119890 + 02

(16229119890 + 02)73894119890 + 01

(36633119890 + 01)46563119890 + 01

(25861)374782119890 + 01

(27904)47741119890 + 01

(106763)

70 11834119890 + 04

(19184119890 + 03)52635119890 + 03

(40246119890 + 03)20414119890 + 02

(74873119890 + 01)75240119890 + 01

(12063119890 + 01)65838119890 + 01

(15847119890 + 01)84068119890 + 01

(30477119890 + 01)

1198916

10 51090(30667)

36806(17685)

38703(17882)

09750(09693)

22685(13272)

06765(08231)

30 15696119890 + 02

(39880119890 + 01)32291119890 + 01

(10059119890 + 01)29949119890 + 01

(71988)17282119890 + 01

(54892)18148119890 + 01

(62546)13312119890 + 01

(44942)

50 38006119890 + 02

(35258119890 + 01)75141119890 + 01

(2587119890 + 01)61896119890 + 01

(15107119890 + 01)36575119890 + 01

(94137)36236119890 + 01

(91136)30739119890 + 01

(85709)

70 58512119890 + 02

(36592119890 + 01)16259119890 + 02

(62498119890 + 01)88994119890 + 01

(17552119890 + 01)54198119890 + 01

(11482119890 + 01)52752119890 + 01

(12361119890 + 01)49043119890 + 01

(11607119890 + 01)

1198917

10 98646119890 minus 05

(00009)00041(00096)

00017(00055)

0(0)

0(0)

0(0)

30 01366(01090)

00008(00045)

00013(00071)

10210119890 minus 10

(10021119890 minus 09)35527119890 minus 17

(14708119890 minus 16)17067119890 minus 11

(13918119890 minus 10)

50 05856(00891)

00027(00021)

99034119890 minus 05

(00009)00001(00007)

24547119890 minus 15

(88945119890 minus 15)36456119890 minus 05

(00001)

70 06941(00646)

00329(00190)

00002(00010)

00011(00017)

44137119890 minus 11

(27956119890 minus 10)00007(00012)

Mathematical Problems in Engineering 7

Table 2 Continued

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198918

10 58516119890 minus 18

(25897119890 minus 17)22299119890 minus 40

(16366119890 minus 39)20832119890 minus 84

(16585119890 minus 83)25229119890 minus 87

(11045119890 minus 86)58546119890 minus 237

(0)22665119890 minus 154

(22664119890 minus 153)

30 19895119890 + 02

(62708119890 + 02)10418119890 minus 06

(37845119890 minus 06)73163119890 minus 18

(27837119890 minus 17)10483119890 minus 39

(72123119890 minus 39)30968119890 minus 99

(30467119890 minus 98)10029119890 minus 41

(10029119890 minus 40)

50 19927119890 + 07

(71603119890 + 07)18348119890 + 01

(48704119890 + 01)00017(00067)

65668119890 minus 25

(31648119890 minus 24)15985119890 minus 47

(14843119890 minus 46)37443119890 minus 23

(37143119890 minus 22)

70 15637119890 + 13

(83656119890 + 13)26886119890 + 08

(16837119890 + 09)14973119890 + 04

(10807119890 + 05)49509119890 minus 19

(26306119890 minus 18)32094119890 minus 29

(25130119890 minus 28)19384119890 minus 19

(10414119890 minus 18)

1198919

10 00472(01618)

01115(02369)

01773(02354)

50448119890 minus 15

(13412119890 minus 15)01269(02045)

00132(00770)

30 15006(02835)

05798(00888)

05469(00765)

03709(00736)

04187(00747)

03822(00767)

50 18625(01067)

06039(01209)

05348(00655)

03691(00544)

03848(00541)

03631(00478)

70 18765(00728)

08228(02349)

04790(00512)

03439(00463)

03369(00469)

03393(00413)

11989110

10 94158119890 minus 13

(49298119890 minus 12)14531119890 minus 26

(39844119890 minus 26)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)25554119890 minus 33

(10557119890 minus 32)

30 00527(00668)

34402119890 minus 07

(54202119890 minus 07)99564119890 minus 15

(20677119890 minus 14)67912119890 minus 12

(46051119890 minus 11)33310119890 minus 31

(14574119890 minus 30)83251119890 minus 13

(42490119890 minus 12)

50 13060(04239)

00010(00008)

18317119890 minus 07

(34350119890 minus 07)00003(00015)

00018(00127)

00011(00091)

70 26961(04601)

00350(00393)

00011(00091)

00059(00115)

00009(00090)

00023(00094)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0Set the coefficients as 119886 = 20 119887 = 02 119888 = 2 sdot 120587

(10) Penalised function one has

11989110 (119909) = 01sin2 (31205871199091) +

119863minus1

sum

119894=1

(119909119894minus 1)2[1 + sin2 (3120587119909

1)]

+ (119909119863minus 1)2[1 + sin2 (2120587119909

119863)]

+

119863

sum

119894=1

119906 (119909119894 5 100 4)

(23)

where

119906 (119909119894 119886 119896 119898) =

119896(119909119894minus 119886)119898 119909

119894gt 119886

0 minus119886 le 119909119894le 119886

119896(minus119909119894minus 119886)119898 119909119894lt minus119886

119910119894= 1 +

1

4

(119909119894+ 1)

(24)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [1 1]119863 with 119891(119909) = 0

42 The Coefficients Setting The parameters for simulationare listed in Table 1

In this table 119888(sdot)

expresses the accelerations coefficients120596 denotes inertia weight the dimension is 119863 and the rangeof the search space and the velocity space are 119909

(sdot)and V(sdot) If

the current position is out of the search space the positionof the particle is taken to be the value of the boundaryand the velocity is taken to be zero If the velocity of theparticle is outside of the boundary its value is set to bethe boundary value The maximum number of iterations isset to 1000 For each function 100 trials were carried outand the average optimal value and the standard deviationare presented To verify the performance of the algorithm atdifferent dimensions variable dimension 119863 increases from10 to 100 and the optimal mean and variance of benchmarkfunctions are calculated The results are presented in Table 2

5 The Results in Comparison withthe Previous Developments

The simulation results are given in Table 2 The comparisonresults elucidate that the searching accuracy and stabilityranging from low to high are listed as BPSO LWPSO EPSOTVAC ECPSO and CPSO for unimodal function It isobvious that the performances of ECPSO and CPSO aresuperior due to their advantage of obtaining the optimalspeed direction and the searching efficiency while in themultimodal function the CPSO algorithm is easy to trap intolocal minimum and TVAC shows better performance thanCPSO Combining the advantages of CPSO and TVAC theEPSO algorithm is applied to enhance the global search in

8 Mathematical Problems in Engineering

0 200 400 600 800 1000

0

05

1

15

2

25

3

35

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

0 200 400 600 800 1000

0

2

4

6

8

10

12

14

16

18

Number of generations

Mea

n be

st va

lue

0 200 400 600 800 1000

0

1

2

3

4

5

Number of generations

Number of generations Number of generations

Number of generations

0 200 400 600 800 1000

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000

0

100

200

300

400

500

0 200 400 600 800 1000

0

10

20

30

40

50

60

minus05

minus10

minus10minus100

minus1

minus2

f1

f5

f4f3

f2

f6

Figure 1 Continued

Mathematical Problems in Engineering 9

0 200 400 600 800 1000

0

005

01

015

02

025

03

035

Number of generations

Number of generations Number of generations

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

0 200 400 600 800 1000

0

05

1

15

2

25

3

0 200 400 600 800 1000

0

05

1

15

2

25

0 200 400 600 800 1000

0

002

004

006

008

01

012

014

016

minus05

minus05

minus005

minus002

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

f8f7

f10f9

Figure 1 Variation of the average optimum value with time

the early part of the optimization and encourage the particlesto converge toward the global optima at the end of the searchComparing to TVAC andCPSO the proposed new algorithmECPSO is appropriate for multimodal function search

With the increase of benchmark functions dimensionthe searching accuracy and stability for each algorithmare decreased The performances of CPSO and ECPSO aresuperior to the other algorithms With regard to multimodalfunction the performances of ECPSOandTVACare superiorto CPSO The average fitness is varied as in Table 2 FromFigure 1 it shows that the order of searching speed fromhigh to low is BPSO (green solid curve) LWPSO (red circle)EPSO (blue triangle) TVAC (black dot curve) CPSO (bluedash-dot) and ECPSO (purples dash) It is obvious that theperformances of EPSO and CPSO are more effective than theother algorithms

6 Calibrate the UnderwaterTransponder Coordinates

After two decades of dedicated research and developmentunmanned underwater vehicles (UUV) have been acceptedby an increasing number of users in bothmilitary and civilianinstitutions The design and implementation of navigationsystems stand out as one of the most critical steps towardsthe successful operation of autonomous vehicles The qualityof the overall estimates of the navigation system dramaticallyinfluences the capability of the vehicles to perform precision-demanding tasks [32] From a navigation point of view asingle range transponder may be regarded as an underwaterlighthouse providingUUVwith the ranges relative to its fixedgeographical location Single transponder navigation is not anew concept As mentioned the first at-sea demonstration of

10 Mathematical Problems in Engineering

Table 3 The calibration of the underwater transponder coordinates

BPSO LWPSO EPSO TVAC CPSO ECPSO1198901 35371119890 minus 15 minus23573119890 minus 15 26542119890 minus 15 minus19813119890 minus 15 minus11700119890 minus 15 14526119890 minus 15

1198902 15510119890 minus 15 23280119890 minus 15 minus48804119890 minus 16 minus14289119890 minus 15 13702119890 minus 15 22549119890 minus 15

1198903 0 0 0 0 0 0119904119902 41948119890 minus 15 27057119890 minus 15 26986119890 minus 15 24428119890 minus 15 18018119890 minus 15 26823119890 minus 15

Layingpoint

z

yG

x

Surfaceship

Transponder

Seabed

Figure 2 The calibration geometry for calibration of the transpon-der position

single transponder UTP aided inertial navigation was carriedout in 2003 as described in [33]

For ranging techniques like UTP to work the geo-graphical location of the transponders must be known Thepreferred method is to measure the position directly usingUSBL on a surface ship When the transponder deploymentis completed a surface ship with USBL and GPS sails aroundthe transponder in a circular motion collecting surface shipposition and distance information The calibration geometryis illustrated in Figure 2

In order to set the design framework let 119866 denote theglobal coordinate frame and let 119861 denote a coordinateframe attached to the vehicle usually denominated as body-fixed coordinate frameThe frame 119874 is the transceiver coor-dinates The position of transponder in the global coordinateframe is given by

119866119883119879=119866119883119863minus

119866

119861119877119861119883119863+119866

119861119877(

119861

119874119877119874119883119879+119861119883119874) (25)

where 119866119883119863is the position of the GPS in global coordinates

119861119883119863is the position of the GPS in vehicle coordinates 119874119883

119879

is the position of the transponder in transceiver coordinates119861119883119874is the position of the transceiver in vehicle coordinates

119866

119861119877is the rotation matrix from 119861 to 119866 and 119861

119874119877is the

rotation matrix from 119874 to 119861After the data collection we can get the distance between

the transponder and transceiver and the corresponding GPSposition in the global coordinates The attitude of the vehiclecan be measured by the heading sensors and the pitchrollsensor so 119866

119861119877is known We also know 119861119883

119863 119861119883119874 and 119861

119874119877

According to (25) we have

119874119883119879=

119861

119874119877minus1(

119866

119861119877minus1(119866119883119879minus119866119883119863) +119861119883119863minus119861119883119874) (26)

We denote 119874119883119879119894= [119874119909119879119894

119874119910119879119894

119874119911119879119894]

119879

(119894 = 1 2 119873)

as the position of the transponder in the transceiver coor-dinates in the 119894th measurement point The cost functionbecomes

119865 =

119873

sum

119894=1

(radic119874

1199092

119879119894+

119874

1199102

119879119894+

119874

1199112

119879119894minus 119871119894)

2

119873

(27)

Particles optimization algorithms have been introducedDefining each particle as a coordinate of the transponderTheparameters of particle swarmoptimization for simulationare the same as in Section 4 And a surface ship moves ina circular motion with a radius of 100 the real coordinatesof transponder are 119866119883

119903119879= [0 0 100]

119879 To simplify theproblem 119866

119861119877is unit matrix and 119861119883

119863minus119861119883119874= [0 0 0]

119879and ignore the sensor measurement error The simulationdata is shown in Table 3 Obviously the particle swarmalgorithm can search for the transponder coordinates andobtain accurate results At the same time CPSO shows thebest performance

Remark 2 One has

1198901 =119866119909119879minus119866119909119903119879 1198902 =

119866119910119879minus119866119910119903119879

1198903 =119866119911119879minus119866119911119903119879 119904119902 = radic1198901

2+ 11989022+ 11989032

(28)

7 Conclusion

In this paper a novel strategy to improve the performance ofparticle swarm optimization is proposed to apply in calibra-tion of the underwater transponder coordinates To improvethe population-based search optimization algorithm eachparticle is evolved along two different directions to generatetwo homologous particles The cost functions of two homol-ogous particles are calculated to keep the optimal one andto eliminate the poor one Then the next generation particleis updated It is regarded as CPSO Ten classify benchmarkfunctions are introduced to reflect the effectiveness of theproposed algorithmThe simulation results demonstrate thatthe unimodal function CPSO algorithm is superior toBPSO LWPSO EPSO and TVAC on the searching accuracystability and convergence speed However considering themultimodal function the performance of TVAC is superiorto CPSO

Mathematical Problems in Engineering 11

Secondly to further improve the performance of CPSOthe ECPSO is proposed by combining the CPSO and theTVAC In the initial period of the evolution the individualexperience is a significant aspect with larger accelerationcoefficient and in the final period the swarm experience issuperior with a greater acceleration coefficient Simultane-ously the evolution for each particle at any time is towardstwo different inertia directions to generate two homologousparticles and to obtain its next generation particles Thesimulations show the effectiveness of multimodal function byusing ECPSOWith the incensement of benchmark functionsdimensions the accuracy and stability of each algorithm willdecrease but CPSO and EPSO display the best performance

At last the strategy to calibrate the underwater transpon-der coordinates using particle swarm algorithm is intro-duced As the cost function for transponder coordinates is aunimodal function CPSO shows better performance than theother algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the Natural Science Foun-dation of China (51179038 1109043 51309067E091002) andthe Program of New Century Excellent Talents in University(NCET-10-0053)

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the 1995 IEEE International Conference on NeuralNetworks vol 4 no 2 pp 1942ndash1948 December 1995

[2] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[3] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[4] M S Arumugam M V C Rao and A W C Tan ldquoA noveland effective particle swarm optimization like algorithm withextrapolation techniquerdquo Applied Soft Computing Journal vol9 no 1 pp 308ndash320 2009

[5] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 1999

[6] G Chen X Huang J Jia et al ldquoNatural exponential inertiaweight strategy in particle swarm optimizationrdquo in Proceedingsof the 6thWorld Congress on Intelligent Control and Automation(WCICA rsquo06) vol 1 pp 3672ndash3675 IEEE June 2006

[7] Q Ni and J Deng ldquoA new logistic dynamic particle swarm opti-mization algorithm based on random topologyrdquo The ScientificWorld Journal vol 2013 Article ID 409167 8 pages 2013

[8] M M Noel ldquoA new gradient based particle swarm optimiza-tion algorithm for accurate computation of global minimumrdquoApplied Soft Computing Journal vol 12 no 1 pp 353ndash359 2012

[9] M G Epitropakis V P Plagianakos and M N VrahatisldquoEvolving cognitive and social experience in particle swarmoptimization through differential evolutionrdquo in Proceedings ofthe IEEE Congress on Evolutionary Computation (CEC rsquo10) pp1ndash8 IEEE July 2010

[10] A A Mousa M A El-Shorbagy and W F Abd-El-WahedldquoLocal search based hybrid particle swarm optimization algo-rithm for multiobjective optimizationrdquo Swarm and Evolution-ary Computation vol 3 pp 1ndash14 2012

[11] M H Moradi and M Abedini ldquoA combination of genetic algo-rithm and particle swarm optimization for optimal DG locationand sizing in distribution systemsrdquo International Journal ofElectrical Power and Energy Systems vol 34 no 1 pp 66ndash742012

[12] W D Chang J P Cheng M C Hsu et al ldquoParameteridentification of nonlinear systems using a particle swarmoptimization approachrdquo in Proceedings of the 3rd InternationalConference on Networking and Computing (ICNC rsquo12) pp 113ndash117 IEEE 2012

[13] J J Soon and K S Low ldquoPhotovoltaic model identificationusing particle swarm optimization with inverse barrier con-straintrdquo IEEE Transactions on Power Electronics vol 27 no 9pp 3975ndash3983 2012

[14] B Jiang N Wang and L Wang ldquoParameter identification forsolid oxide fuel cells using cooperative barebone particle swarmoptimization with hybrid learningrdquo International Journal ofHydrogen Energy vol 39 no 1 pp 532ndash542 2013

[15] A Alfi ldquoParticle swarm optimization algorithm with DynamicInertia Weight for online parameter identification applied toLorenz chaotic systemrdquo International Journal of InnovativeComputing Information and Control vol 8 no 2 pp 1191ndash12042012

[16] X Hu S Shi and X Gu ldquoAn improved particle swarm opti-mization algorithm for wireless sensor networks localizationrdquoin Proceedings of the 8th International Conference on WirelessCommunications Networking and Mobile Computing (WiCOMrsquo12) pp 1ndash4 Shanghai China September 2012

[17] F Sun J Yu and D Xu ldquoVisual measurement and control forunderwater robots a surveyrdquo in Proceedings of the 25th ChineseControl andDecision Conference (CCDC rsquo13) pp 333ndash338 IEEE2013

[18] V Djapic D Nad G Ferri et al ldquoNovel method for underwaternavigation aiding using a companion underwater robot asa guiding platformsrdquo in Proceedings of the 2013 MTSIEEEOCEANS-Bergen pp 1ndash10 IEEE June 2013

[19] A Caiti V Calabro D Meucci et al ldquoUnderwater Robots PastPresent and Futurerdquo 422ndash437

[20] P Krishnamurthy and F Khorrami ldquoA self-aligning underwaternavigation system based on fusion ofmultiple sensors includingDVL and IMUrdquo in Proceedings of the 9th Asian Control Confer-ence (ASCC rsquo13) pp 1ndash6 IEEE 2013

[21] S Wang and F Kang ldquoResearch and design of UUV navigationand control integrative simulation system based on compo-nentrdquo Intelligent Information Management vol 4 no 5 pp 181ndash187 2012

[22] T Kashima A Asada and T Ura ldquoThe positioning sys-tem integrated LBL and SSBL using seafloor acoustic mirrortransponderrdquo in Proceedings of the IEEE International Underwa-ter Technology Symposium (UT rsquo13) March 2013

12 Mathematical Problems in Engineering

[23] P Batista C Silvestre and P Oliveira ldquoGAS tightly coupledLBLUSBL position and velocity filter for underwater vehiclesrdquoin Proceedings of the European Control Conference (ECC rsquo13) pp2982ndash2987 Zurich Switzerland 2013

[24] M Morgado P Oliveira and C Silvestre ldquoTightly coupledultrashort baseline and inertial navigation system for under-water vehicles an experimental validationrdquo Journal of FieldRobotics vol 30 no 1 pp 142ndash170 2013

[25] Y U Min and H Junyin ldquoThe calibration of the USBL trans-ducer array for Long-range precision underwater positioningrdquoin Proceedings of the IEEE 10th International Conference onSignal Processing (ICSP rsquo10) pp 2357ndash2360 IEEE October 2010

[26] D Ji Y Li and J Liu ldquoSeafloor transponder calibration usingimproved perpendiculars intersectionrdquoAppliedOceanResearchvol 32 no 3 pp 261ndash266 2010

[27] B Jiao Z Lian and Q Chen ldquoA dynamic global and local com-bined particle swarm optimization algorithmrdquo Chaos Solitonsand Fractals vol 42 no 5 pp 2688ndash2695 2009

[28] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 IEEE May 1998

[29] M Schwaab E C Biscaia Jr J L Monteiro and J CPinto ldquoNonlinear parameter estimation through particle swarmoptimizationrdquo Chemical Engineering Science vol 63 no 6 pp1542ndash1552 2008

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII pp591ndash600 Springer Berlin Germany 1998

[31] A Chander A Chatterjee and P Siarry ldquoA new social andmomentum component adaptive PSO algorithm for imagesegmentationrdquo Expert Systems with Applications vol 38 no 5pp 4998ndash5004 2011

[32] M Morgado P Batista P Oliveira and C Silvestre ldquoPositionUSBLDVL sensor-based navigation filter in the presence ofunknown ocean currentsrdquoAutomatica vol 47 no 12 pp 2604ndash2614 2011

[33] B Jalving K Gade O K Hagen and K Vestgard ldquoA toolboxof aiding techniques for the HUGIN AUV integrated inertialnavigation systemrdquo in Proceedings of the OCEANS 2003 vol 2pp 1146ndash1153 IEEE September 2003

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

evolution algorithm In an attempt to efficiently guide theevolution and enhance the convergence the author evolvedthe personal experience or memory of the particles withthe differential evolution algorithm without destroying thesearch capabilities of the algorithmMousa et al [10] proposea hybrid multiobjective evolutionary algorithm combininggenetic algorithm and particle swarm optimization the localsearch scheme is implemented as a neighborhood searchengine to improve the solution quality where it intendsto explore the less-crowded area in the current archive topossibly obtain more nondominated solutions

As a kind of optimization algorithm PSO is simple instructure has good performance and is easy to implement Itis widely applied in various engineering applications Moradiand Abedini [11] combined genetic algorithm and particleswarm optimization for optimal location and sizing of dis-tributed generation on distribution systemsThe algorithm isto minimize network power losses make better voltage regu-lation and improve the voltage stabilitywithin the frameworkof system operation and security constraints in radial distri-bution systems Chang et al [12] apply the PSO algorithmto estimate the parameters of the Genesio-Tesi nonlinearchaotic systems and the estimation of the PSO algorithmis verified by examining different sets of random initialpopulations under the presence of measurement noise Soonand Low [13] proposed a new approach using particle swarmoptimization with inverse barrier constraint to determinethe unknown photovoltaic model parameters The proposedmethod has been validated with three different photovoltaictechnologies Jiang et al [14] proposed the barebone particleswarm optimization algorithm to determine the parametersof solid oxide fuel cell (SOPC) The cooperative coevolutionstrategy is applied to divide the output voltage functioninto four subfunctions based on the interdependence amongvariables To the nonlinear characteristic of SOPC model ahybrid learning strategy is proposed for BPSO to ensure agood balance between exploration and exploitation Alfi [15]proposed novel particle swarm optimization to cope with theonline system parameter identification problem The inertiaweight for every particle is dynamically updated based on thefeedback taken from the fitness of the best previous positionfound by the particle and a novel methodology is incorpo-rated into the novel particle swarm optimization to be ableto effectively response and detect any parameter variationsof system to be identified Hu and Shi [16] to solve the pre-mature convergence problem of PSO improved algorithmswith hybrid and mutation operators leading to obtaininga high level of particle population diversity decreasing thepossibility of falling into local optima and improving locationaccuracy The novel algorithm is introduced in the range-based location for wireless sensor networks and simulationshows a better performance than basic PSO algorithm

With the development of marine economy and technol-ogy unmanned underwater vehicle (UUV) is an effectivemeans for marine detection resource exploitation militaryinterfere and investigation [17ndash19] Navigation of UUVhas been and remains a substantial challenge to platformsOne of the main driving factors is the ability to carryout long-duration missions fully autonomously and without

supervision from a surface ship [20 21] Combined withinertial navigation the use of one or several transponderson the seabed is an accurate and cost-effective approachtoward solving several of these challenges [22ndash24] It isobvious that the exact position of the transponder is veryimportant in the underwater transponder positioning system[25 26] However in the practical operations due to theinfluence of ocean currents and other factors the practicalcoordinates of transponder will drift from the position whereit launched into the water So it requires the mother shipto calibrate the coordinate of the transponder this paperproposed the particle swarm optimization algorithm solvingthe transponder coordinates

The contribution of this paper is concluded as the fol-lowing Firstly considering the competition particle swarmalgorithm each particle will evolve along two differentdirections to generate two homologous particlesThe optimalone is kept through comparing the cost functions of twohomologous particles and the next generation particle willbe obtained finally Secondly according to the advantage ofTVAC combining CPSO and TVAC ECPSO algorithm ispresentedWith a large cognitive component and small socialcomponent at the beginning on the other hand a smallcognitive component and a large social component allow theparticles to converge to the global optima in the latter partof the optimization Simultaneously the evolution for eachparticle at any time is along two different inertia directionsto generate two homologous particles and to obtain nextgeneration particle Lastly ECPSO is introduced to calibratethe coordinate of the transponder

The rest of this paper is organized as follows In Section 2the basic PSOand its previous developments are summarizedIn Section 3 the competition particle swarm optimizationalgorithm and extension competition particle swarm opti-mization algorithm are introduced The experimental set-tings for the benchmark functions and simulation strategiesare explained and the conclusion is drawn based on thecomparison analysis In Section 4 ECPSO is introduced tocalibrate the coordinates of the transponder and simulationsare designed to verify the feasibility of the algorithm present

2 Some Previous Work

Introduced by Dr Kennedy and Dr Eberhart in 1995 PSOhas ever since turned out to be a competitor in the field ofnumerical optimization and there has been a considerableamount of work done in developing the original version ofPSO In this section we summarize some entire significantprevious developments

21 Basic Particle Swarm Optimization (BPSO) In PSO eachsolution called a ldquoparticlerdquo flies in the search space searchingfor the optimal position to land PSO system combineslocal search method (through individual experience) withglobal search methods (through neighboring experience)attempting to balance exploration and exploitation [27] Eachparticle has a position vector 119909

119894(119896) a velocity vector V

119894(119896) the

position with the best fitness encountered by the particle and

Mathematical Problems in Engineering 3

the index of the best particle in the swarmTheposition vectorand the velocity vector of the 119894th particle in the119889-dimensionalsearch space can be represented as 119909

119894= (1199091198941 1199091198942 1199091198943 119909

119894119889)

and V119894= (V1198941 V1198942 V1198943 V

119894119889) respectively The best position

of each particle (119901best) is 119901119894= (1199011198941 1199011198942 1199011198943 119901

119894119889) and

the fitness particle found so far at generation 119896 (119892best) is119901119892= (1199011198921 1199011198922 119901

119892119889) In each generation each particle is

updated by the following two equations

V119894119889 (119896 + 1) = V

119894119889 (119896) + 1198881

times 1199031times (119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 1198882times 1199032times (119901119892119889 (

119896) minus 119909119894119889 (119896))

(1)

119909119894119889 (119896 + 1) = 119909119894119889 (

119896) + V119894119889 (119896 + 1) (2)

The parameters 1198881and 1198882are constants known as acceler-

ation coefficients 1199031and 1199032are random values in the range

from 0 to 1 and the value of 1199031and 1199032is not the same for

every iteration Kennedy and Eberhart [1] suggested settingeither of the acceleration coefficients at 2 in order tomake themean of both stochastic factors in (1) unity so that particleswould over fly only half the time of search The first equationshows that in PSO the search toward the optimum solutionis guided by the previous velocity the cognitive componentand the social component

Since the introduction of the particle swarm optimiza-tion numerous variations of the algorithm have been devel-oped in the literature Eberhart and Shi showed that PSOsearches for wide areas effectively but tends to lack localsearch precision They proposed in that work a solution byintroducing 120596 an inertia factor In this paper we name it asbasic particle swarm optimization (BPSO)

V119894119889 (119896 + 1) = 120596 times V

119894119889 (119896) + 1198881

times 1199031times (119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 1198882times 1199032times (119901119892119889 (

119896) minus 119909119894119889 (119896))

119909119894119889 (119896 + 1) = 119909119894119889 (

119896) + V119894119889 (119896 + 1)

(3)

22 Time-Varying Inertia Weight (TVIW) The role of theinertia weight 120596 is considered very important in PSO conver-gence behavior The inertia weight is applied to control theimpact of the previous history of velocities on the currentvelocity large inertia weight facilitates global explorationwhile small one tends to facilitate local exploration In orderto assure that the particles converge to the best point inthe course of the search Shi and Eberhart [28] have foundthat time-varying inertia weight (TVIW) has a significantimprovement in the performance of PSO and proposedlinear decreasing inertia weight PSO (LWPSO) with a lineardecreasing value of 120596 This modification can increase theexploration of the parameter space during the initial searchiterations and increase the exploitation of the parameter spaceduring the final steps of the search [29] The mathematicalrepresentation of inertia weight is given as follows

120596 = (1205961minus 1205962) times (

MAXITER minus 119896MAXITER

) + 1205962 (4)

where 1205961and 120596

2are the initial and final values of the inertia

weight respectively 119896 is the current iteration number and

MAXITER is the maximum number of allowable iterationsShi and Eberhart [5] indicate that the optimal solution can beimproved by varying the value of 120596 from 09 at the beginningof the search to 04 at the end of the search formost problems

Chen et al [6] proposed natural exponential (base 119890)inertia weight strategies named EPSO and expressed as

120596 = 1205962+ (1205961minus 1205962) times exp[minus( 119870

(MAXITER4))

2

] (5)

23 Time-Varying Acceleration Coefficient (TVAC) In PSOthe particle was updated due to the cognitive component andthe social component Therefore proper control of these twocomponents is very important to find the optimum solutionaccurately and efficiently Ratnaweera et al [2] introduced atime-varying acceleration coefficient (TVAC) which reducesthe cognitive component and increases the social componentby changing the acceleration coefficients 119888

1and 119888

2with

the time evolution The objective of this development is toenhance the global search in the early part of the optimizationand to encourage the particles to converge toward the globaloptima at the end of the search The TVAC is representedusing the following equations

1198881= (119888max minus 119888min)

119896

MAXITR+ 119888min

1198882= (119888min minus 119888max)

119896

MAXITR+ 119888max

(6)

where 119888min and 119888max are constants 119896 is the current iterationnumber and MAXITR is the maximum number of allowableiterations

Simulations were carried out with numerical bench-marks to find out the best ranges of values for 119888

1and 1198882

From the results it was observed that the best solutions weredetermined when changing 119888

1from 25 to 05 and changing 119888

2

from 05 to 25 over the full search range

3 Proposed New Developments

It is clarified from (1) that particlersquos new velocity is correlatedwith three terms the particlersquos previous velocity the valueof the cognitive component and the value of the socialcomponent Therefore proper control method with inertiaweight factor and acceleration coefficients is significant tofind the optimum solution accurately and efficiently

The inertia weight is utilized to adjust the influence of theprevious velocity on the current velocity and balance betweenglobal and local exploration abilities of the ldquoflying particlerdquo[30 31] A larger inertia weight implies stronger globalexploration ability advocating the particle to escape from alocal minimum A smaller inertia weight leads to strongerlocal exploration ability confining the particle searchingwithin a local range near its present position to guarantee theconvergence

Kennedy and Eberhart [1] indicated that a relatively highvalue of the cognitive component compared with the socialcomponent will result in excessive wandering of individuals

4 Mathematical Problems in Engineering

through the search space In contrast a relatively high value ofthe social component may lead particles to rush prematurelytoward a local optimum

Considering those concerns we propose a new strategyfor the PSO concept

31 Competition Particle Swarm Optimization (CPSO) Inthe process of particle evolution each particle is evolvedalong different directions with different inertia coefficientsand acceleration coefficients Two homologous particles aregenerated and the optimal one is kept through comparingcost functions of two homologous particles eliminating theinferior one Then the next generation particle is updatedfinally The evaluation function of each particle is describedas

V119905119894119889(119896 + 1) = 120596

119905V119894119889 (119896) + 119888

119905

1119903119905

1(119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 119888119905

2119903119905

2(119901119892119889 (

119896) minus 119909119894119889 (119896))

119909119905

119894119889(119896 + 1) = 119909

119905

119894119889(119896) + V119905

119894119889(119896 + 1)

(7)

And the final equations are shown as

V119894119889 (119896) = V119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

119909119894119889 (119896) = 119909

119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

(8)

where 119899 is the number of particles in the swarm 119872 is themaximum iteration frequency 119903119905

1 1199031199052are the randomnumbers

in the range of [0 1] 119909119894119889(119896) is the 119894th particle position in the

119889th dimension after 119896 time iteration 119909119905119894119889(119896) shows the 119889th

dimension position for subparticle 119905 of the 119894th particle after119896 time iteration 120596119905 is the speed inertia weight of subparticle119905 1198881199051and 1198881199052are constants denoting acceleration coefficients

fitness (119909119905119894(119870)) is the fitting function of the subparticle 119905 after

119896 time iteration V119905119894119889(119896) is the speed of subparticle 119905 at 119889th

dimension 119901119894119889(119896) is the optimal position of the 119894th particle

at 119889th dimension and 119901119892119889(119896) is the swarm optimal position

at 119889th dimension after 119896 time iteration

Remark 1 In this paper two subparticles are generated foreach particle at one time therefore 119905 = 1 2

The detailed steps are shown as follows

Step 1 Initial

Substep 1 Set the initial parameters 119899119872 119908119905 1198881199051 1198881199052

Substep 2 Take random initial 119909119894119889(0)

Substep 3 Take random initial V119905119894119889(1)

Substep 4 Calculate fitness(119909119905119894(0)) and set 119901

119894(0) = 119909

119894(0)

Substep 5 One has 119901119892(0) = 119909

119894(0) | 119909

119894(0) isin

min(fitness(119909119894(0)))

Step 2 If the criteria are satisfied output the best solutionotherwise go to Substep 6

Substep 6 Update V119905119894119889(119896) and 119909119905

119894119889(119896)

Substep 7 Calculate fitness(119909119905119894(119896))

Substep 8 One has 119909119894119889(119896) = 119909

119905

119894119889(119896) | min(fitness(119909119905

119894(119896))) 119905 =

1 2

Substep 9 If fitness(119909119894(119896)) lt fitness(119901

119894(119896))

119901119894 (119896) = 119909119894 (

119896) (9)

If minfitness(119909119894(119896)) 119894 = 1 119899 lt fitness(119901

119892(119896))

119901119892 (119896) = 119909119894 (

119896) | min [fitness (119909119894 (119896)) 119894 = 1 119899] (10)

Substep 10 Go back to Step 2

32 Extension Competition Particle Swarm Optimization(ECPSO) competition particle swarm optimization (CPSO)helps adjust the search direction particles and improve thesearch speed and efficiency but due to rapid convergenceCPSO is easy to fall into local minima According tobenchmark functions simulation in Section 4 it is obviousthat CPSO is superior to the searching effective of single-modulus function and TVAC is superior to the searchingeffective of multimodulus function The reason resulting inthis phenomenon is due to the selection of accelerationcoefficient With a large cognitive component and a smallsocial component at the beginning particles are allowed tomove around the search space instead of moving towardthe population best On the other hand a small cognitivecomponent and a large social component allow the particlesto converge to the global optima in the latter part of theoptimization Considering the advantage of TVAC introduceTVAC into CPSO and the extension competition particleswarm optimization (ECPSO) with the acceleration coeffi-cients proposed as above The evolution equations can bemathematically represented as the following

V119905119894119889(119896 + 1) = 120596

119905V119894119889 (119896) + 1198881

119903119905

1(119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 1198882119903119905

2(119901119892119889 (

119896) minus 119909119894119889 (119896))

119909119905

119894119889(119896 + 1) = 119909

119905

119894119889(119896) + V119905

119894119889(119896 + 1)

(11)

where

1198881= (119888max minus 119888min)

119896

MAXITR+ 119888min

1198882= (119888min minus 119888max)

119896

MAXITR+ 119888max

(12)

And it is obvious that

V119894119889 (119896) = V119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

119909119894119889 (119896) = 119909

119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

(13)

Mathematical Problems in Engineering 5

4 Experimental Settings and SimulationStrategies for Benchmark Testing

Simulations were carried out to observe the rate of con-vergence and the quality of the optimum solution of thenew methods introduced in this investigation by comparingwith BPSO EPSO and TVAC From the standard set ofbenchmark problems available in the literature there are 5important functions considered to test the efficacy of theproposed method All of the benchmark functions reflectdifferent degrees of complexity

41 Functions Introduction The functions are as follows

(1) Sphere function one has

1198911 (119909) =

119863

sum

119894=1

1199092

119894 (14)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is verysimple convex and unimodal function with only one localoptimum value

(2) Axis parallel hyperellipsoid function one has

1198912 (119909) =

119863

sum

119894=1

119894 sdot 1199092

119894 (15)

It is known as the weighted sphere model With the searchspace 119909

119894| minus100 lt 119909

119894lt 100 the global minimum locates at

119909 = [0 0]119863 with 119891(119909) = 0 It is continuous convex and

unimodal

(3) Rotated hyperellipsoid function (Schwefelrsquos problem12) one has

1198913 (119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

(16)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is

continuous convex and unimodal With respect to the coor-dinate axes this function produces rotated hyperellipsoids

(4) Moved axis parallel hyperellipsoid function one has

1198914 (119909) =

119863

sum

119894=1

5119894 sdot 1199092

119894 (17)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909(119894) = 5 lowast 119894 119894 = 1 119863 with 119891(119909) = 0

(5) Rosenbrock function one has

1198915 (119909) =

119863minus1

sum

119894=1

[100(119909119894+1minus 1199092

119894)

2

+ (119909119894minus 1)2] (18)

Table 1 Parameters for simulation

Method Parameters

CPSO1205961= 120596max = 09 120596

2= 120596min = 04

1198881

1= 05 1198881

2= 25 1198882

1= 25

1198882

2= 05

119899 = 30119863 = 10

119909119894isin [minus100 100]

V119894isin [minus100 100]

ECPSO1205961= 120596max = 09 120596

2= 120596min = 04

119888min = 05 119888max = 25

BPSO 1198881= 1198882= 20 120596 = 07

EPSO1198881= 1198882= 20 120596max = 09120596min = 04

TVAC 119888min = 05 119888max = 25 120596 = 07

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [1 1]119863 with 119891(119909) = 0 It

is a unimodal function and the global optimum is inside along narrow parabolic shaped flat valley To find the valley istrivial

(6) Rastrigin function one has

1198916 (119909) =

119863

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894) + 10] (19)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is

highly multimodal However the locations of the minima areregularly distributed

(7) Griewank function one has

1198917 (119909) =

1

4000

119863

sum

119894=1

1199092

119894minus

119863

prod

119894=1

cos(119909119894

radic119894

) + 1 (20)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It

is a multimodal function and has many widespread localminima However the locations of the minima are regularlydistributed

(8) Sum of different power function one has

1198918 (119909) =

119863

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816

(119894+1) (21)

The sum of different powers is a commonly used uni-modal test function With the search space 119909

119894| minus100 lt

119909119894lt 100 the global minimum locates at 119909 = [0 0]

119863

with 119891(119909) = 0

(9) Ackleyrsquos path function one has

1198919 (119909) = minus119886 sdot 119890

minus119887sdotradicsum119863

119894=11199092

119894119863minus 119890sum119863

119894=1cos(119888sdot119909119894)119863

+ 119886 + 1198901

(22)

6 Mathematical Problems in Engineering

Table 2 Comparison of BPSO WPSO EPSO TVAC and CPSO

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198911

10 47979119890 minus 11

(11343119890 minus 10)32219119890 minus 24

(10706119890 minus 23)87747119890 minus 51

(3408119890 minus 50)14826119890 minus 54

(14770119890 minus 53)41023119890 minus 146

(36388119890 minus 145)41221119890 minus 118

(30916119890 minus 117)

30 53768(43459)

40137119890 minus 05

(5926119890 minus 05)23219119890 minus 12

(11495119890 minus 11)11062119890 minus 10

(73302119890 minus 10)14872119890 minus 46

(64922119890 minus 46)16829119890 minus 10

(10243119890 minus 09)

50 28412119890 + 01

(45047)02149(01634)

22215119890 minus 05

(35696119890 minus 05)00019(00079)

93999119890 minus 20

(40558119890 minus 19)00022(00083)

70 44686119890 + 01

(43454)11728119890 + 01

(12113119890 + 01)00192(00238)

00609(01033)

20133119890 minus 07

(19424119890 minus 06)00302(00458)

1198912

10 37589119890 minus 10

(17479119890 minus 09)85629119890 minus 24

(19031119890 minus 23)52910119890 minus 50

(23037119890 minus 49)15947119890 minus 52

(14618119890 minus 51)89632119890 minus 146

(35734119890 minus 145)30640119890 minus 113

(30622119890 minus 112)

30 378269(313869)

00005(00007)

10278119890 minus 11

(20979119890 minus 11)11937119890 minus 09

(65345119890 minus 09)81119119890 minus 45

(44882119890 minus 44)30047119890 minus 10

(21285119890 minus 09)

50 68495119890 + 02

(10535119890 + 02)28041(22863)

00004(00010)

01214(06006)

24666119890 minus 15

(24614119890 minus 14)00223(00835)

70 14952119890 + 03

(18073119890 + 02)12961119890 + 02

(14505119890 + 02)02601(02405)

21146(39189)

12547119890 minus 05

(00001)11879(20240)

1198913

10 00013(00018)

40624119890 minus 08

(11066119890 minus 07)31542119890 minus 15

(19686119890 minus 14)36072119890 minus 27

(16376119890 minus 26)60846119890 minus 60

(58552119890 minus 59)44092119890 minus 51

(31687119890 minus 50)

30 20647119890 + 01

(79831)37596(20255)

05809(03373)

00091(00197)

27957119890 minus 06

(50455119890 minus 06)00258(00385)

50 72983119890 + 01

(28263119890 + 01)26189119890 + 01

(13125119890 + 01)10295119890 + 01

(51265)04883(03021)

00775(00620)

05720(03509)

70 14821119890 + 02

(46282119890 + 01)69337119890 + 01

(33718119890 + 01)29336119890 + 01

(10417119890 + 01)27039(10853)

11214(04917)

22342(07423)

1198914

10 27409119890 minus 09

(15090119890 minus 08)13708119890 minus 22

(65937119890 minus 22)35012119890 minus 49

(28064119890 minus 48)99156119890 minus 52

(69209119890 minus 51)16140119890 minus 144

(14296119890 minus 143)30561119890 minus 115

(21383119890 minus 114)

30 19825119890 + 02

(17202119890 + 02)00025(00029)

73319119890 minus 11

(23165119890 minus 10)66962119890 minus 09

(36744119890 minus 08)10828119890 minus 42

(89052119890 minus 42)56960119890 minus 08

(39998119890 minus 07)

50 32922119890 + 03

(56799119890 + 02)13397119890 + 01

(94756)00015(00019)

03064(12614)

22477119890 minus 17

(14523119890 minus 16)02379(11928)

70 74756119890 + 03

(75131119890 + 02)68348119890 + 02

(61444119890 + 02)16799(29749)

10304119890 + 01

(14839119890 + 01)19407119890 minus 06

(96092119890 minus 06)55072

(10286119890 + 01)

1198915

10 56418(1092)

43037(12401)

31701(12637)

07262(09490)

06503(14786)

06445(14706)

30 12322119890 + 03

(94694119890 + 02)42313119890 + 01

(26596119890 + 01)31052119890 + 01

(17699119890 + 01)23370119890 + 01

(19238)14702119890 + 01

(31506)19234119890 + 01

(29573)

50 65575119890 + 03

(13919119890 + 03)29055119890 + 02

(16229119890 + 02)73894119890 + 01

(36633119890 + 01)46563119890 + 01

(25861)374782119890 + 01

(27904)47741119890 + 01

(106763)

70 11834119890 + 04

(19184119890 + 03)52635119890 + 03

(40246119890 + 03)20414119890 + 02

(74873119890 + 01)75240119890 + 01

(12063119890 + 01)65838119890 + 01

(15847119890 + 01)84068119890 + 01

(30477119890 + 01)

1198916

10 51090(30667)

36806(17685)

38703(17882)

09750(09693)

22685(13272)

06765(08231)

30 15696119890 + 02

(39880119890 + 01)32291119890 + 01

(10059119890 + 01)29949119890 + 01

(71988)17282119890 + 01

(54892)18148119890 + 01

(62546)13312119890 + 01

(44942)

50 38006119890 + 02

(35258119890 + 01)75141119890 + 01

(2587119890 + 01)61896119890 + 01

(15107119890 + 01)36575119890 + 01

(94137)36236119890 + 01

(91136)30739119890 + 01

(85709)

70 58512119890 + 02

(36592119890 + 01)16259119890 + 02

(62498119890 + 01)88994119890 + 01

(17552119890 + 01)54198119890 + 01

(11482119890 + 01)52752119890 + 01

(12361119890 + 01)49043119890 + 01

(11607119890 + 01)

1198917

10 98646119890 minus 05

(00009)00041(00096)

00017(00055)

0(0)

0(0)

0(0)

30 01366(01090)

00008(00045)

00013(00071)

10210119890 minus 10

(10021119890 minus 09)35527119890 minus 17

(14708119890 minus 16)17067119890 minus 11

(13918119890 minus 10)

50 05856(00891)

00027(00021)

99034119890 minus 05

(00009)00001(00007)

24547119890 minus 15

(88945119890 minus 15)36456119890 minus 05

(00001)

70 06941(00646)

00329(00190)

00002(00010)

00011(00017)

44137119890 minus 11

(27956119890 minus 10)00007(00012)

Mathematical Problems in Engineering 7

Table 2 Continued

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198918

10 58516119890 minus 18

(25897119890 minus 17)22299119890 minus 40

(16366119890 minus 39)20832119890 minus 84

(16585119890 minus 83)25229119890 minus 87

(11045119890 minus 86)58546119890 minus 237

(0)22665119890 minus 154

(22664119890 minus 153)

30 19895119890 + 02

(62708119890 + 02)10418119890 minus 06

(37845119890 minus 06)73163119890 minus 18

(27837119890 minus 17)10483119890 minus 39

(72123119890 minus 39)30968119890 minus 99

(30467119890 minus 98)10029119890 minus 41

(10029119890 minus 40)

50 19927119890 + 07

(71603119890 + 07)18348119890 + 01

(48704119890 + 01)00017(00067)

65668119890 minus 25

(31648119890 minus 24)15985119890 minus 47

(14843119890 minus 46)37443119890 minus 23

(37143119890 minus 22)

70 15637119890 + 13

(83656119890 + 13)26886119890 + 08

(16837119890 + 09)14973119890 + 04

(10807119890 + 05)49509119890 minus 19

(26306119890 minus 18)32094119890 minus 29

(25130119890 minus 28)19384119890 minus 19

(10414119890 minus 18)

1198919

10 00472(01618)

01115(02369)

01773(02354)

50448119890 minus 15

(13412119890 minus 15)01269(02045)

00132(00770)

30 15006(02835)

05798(00888)

05469(00765)

03709(00736)

04187(00747)

03822(00767)

50 18625(01067)

06039(01209)

05348(00655)

03691(00544)

03848(00541)

03631(00478)

70 18765(00728)

08228(02349)

04790(00512)

03439(00463)

03369(00469)

03393(00413)

11989110

10 94158119890 minus 13

(49298119890 minus 12)14531119890 minus 26

(39844119890 minus 26)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)25554119890 minus 33

(10557119890 minus 32)

30 00527(00668)

34402119890 minus 07

(54202119890 minus 07)99564119890 minus 15

(20677119890 minus 14)67912119890 minus 12

(46051119890 minus 11)33310119890 minus 31

(14574119890 minus 30)83251119890 minus 13

(42490119890 minus 12)

50 13060(04239)

00010(00008)

18317119890 minus 07

(34350119890 minus 07)00003(00015)

00018(00127)

00011(00091)

70 26961(04601)

00350(00393)

00011(00091)

00059(00115)

00009(00090)

00023(00094)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0Set the coefficients as 119886 = 20 119887 = 02 119888 = 2 sdot 120587

(10) Penalised function one has

11989110 (119909) = 01sin2 (31205871199091) +

119863minus1

sum

119894=1

(119909119894minus 1)2[1 + sin2 (3120587119909

1)]

+ (119909119863minus 1)2[1 + sin2 (2120587119909

119863)]

+

119863

sum

119894=1

119906 (119909119894 5 100 4)

(23)

where

119906 (119909119894 119886 119896 119898) =

119896(119909119894minus 119886)119898 119909

119894gt 119886

0 minus119886 le 119909119894le 119886

119896(minus119909119894minus 119886)119898 119909119894lt minus119886

119910119894= 1 +

1

4

(119909119894+ 1)

(24)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [1 1]119863 with 119891(119909) = 0

42 The Coefficients Setting The parameters for simulationare listed in Table 1

In this table 119888(sdot)

expresses the accelerations coefficients120596 denotes inertia weight the dimension is 119863 and the rangeof the search space and the velocity space are 119909

(sdot)and V(sdot) If

the current position is out of the search space the positionof the particle is taken to be the value of the boundaryand the velocity is taken to be zero If the velocity of theparticle is outside of the boundary its value is set to bethe boundary value The maximum number of iterations isset to 1000 For each function 100 trials were carried outand the average optimal value and the standard deviationare presented To verify the performance of the algorithm atdifferent dimensions variable dimension 119863 increases from10 to 100 and the optimal mean and variance of benchmarkfunctions are calculated The results are presented in Table 2

5 The Results in Comparison withthe Previous Developments

The simulation results are given in Table 2 The comparisonresults elucidate that the searching accuracy and stabilityranging from low to high are listed as BPSO LWPSO EPSOTVAC ECPSO and CPSO for unimodal function It isobvious that the performances of ECPSO and CPSO aresuperior due to their advantage of obtaining the optimalspeed direction and the searching efficiency while in themultimodal function the CPSO algorithm is easy to trap intolocal minimum and TVAC shows better performance thanCPSO Combining the advantages of CPSO and TVAC theEPSO algorithm is applied to enhance the global search in

8 Mathematical Problems in Engineering

0 200 400 600 800 1000

0

05

1

15

2

25

3

35

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

0 200 400 600 800 1000

0

2

4

6

8

10

12

14

16

18

Number of generations

Mea

n be

st va

lue

0 200 400 600 800 1000

0

1

2

3

4

5

Number of generations

Number of generations Number of generations

Number of generations

0 200 400 600 800 1000

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000

0

100

200

300

400

500

0 200 400 600 800 1000

0

10

20

30

40

50

60

minus05

minus10

minus10minus100

minus1

minus2

f1

f5

f4f3

f2

f6

Figure 1 Continued

Mathematical Problems in Engineering 9

0 200 400 600 800 1000

0

005

01

015

02

025

03

035

Number of generations

Number of generations Number of generations

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

0 200 400 600 800 1000

0

05

1

15

2

25

3

0 200 400 600 800 1000

0

05

1

15

2

25

0 200 400 600 800 1000

0

002

004

006

008

01

012

014

016

minus05

minus05

minus005

minus002

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

f8f7

f10f9

Figure 1 Variation of the average optimum value with time

the early part of the optimization and encourage the particlesto converge toward the global optima at the end of the searchComparing to TVAC andCPSO the proposed new algorithmECPSO is appropriate for multimodal function search

With the increase of benchmark functions dimensionthe searching accuracy and stability for each algorithmare decreased The performances of CPSO and ECPSO aresuperior to the other algorithms With regard to multimodalfunction the performances of ECPSOandTVACare superiorto CPSO The average fitness is varied as in Table 2 FromFigure 1 it shows that the order of searching speed fromhigh to low is BPSO (green solid curve) LWPSO (red circle)EPSO (blue triangle) TVAC (black dot curve) CPSO (bluedash-dot) and ECPSO (purples dash) It is obvious that theperformances of EPSO and CPSO are more effective than theother algorithms

6 Calibrate the UnderwaterTransponder Coordinates

After two decades of dedicated research and developmentunmanned underwater vehicles (UUV) have been acceptedby an increasing number of users in bothmilitary and civilianinstitutions The design and implementation of navigationsystems stand out as one of the most critical steps towardsthe successful operation of autonomous vehicles The qualityof the overall estimates of the navigation system dramaticallyinfluences the capability of the vehicles to perform precision-demanding tasks [32] From a navigation point of view asingle range transponder may be regarded as an underwaterlighthouse providingUUVwith the ranges relative to its fixedgeographical location Single transponder navigation is not anew concept As mentioned the first at-sea demonstration of

10 Mathematical Problems in Engineering

Table 3 The calibration of the underwater transponder coordinates

BPSO LWPSO EPSO TVAC CPSO ECPSO1198901 35371119890 minus 15 minus23573119890 minus 15 26542119890 minus 15 minus19813119890 minus 15 minus11700119890 minus 15 14526119890 minus 15

1198902 15510119890 minus 15 23280119890 minus 15 minus48804119890 minus 16 minus14289119890 minus 15 13702119890 minus 15 22549119890 minus 15

1198903 0 0 0 0 0 0119904119902 41948119890 minus 15 27057119890 minus 15 26986119890 minus 15 24428119890 minus 15 18018119890 minus 15 26823119890 minus 15

Layingpoint

z

yG

x

Surfaceship

Transponder

Seabed

Figure 2 The calibration geometry for calibration of the transpon-der position

single transponder UTP aided inertial navigation was carriedout in 2003 as described in [33]

For ranging techniques like UTP to work the geo-graphical location of the transponders must be known Thepreferred method is to measure the position directly usingUSBL on a surface ship When the transponder deploymentis completed a surface ship with USBL and GPS sails aroundthe transponder in a circular motion collecting surface shipposition and distance information The calibration geometryis illustrated in Figure 2

In order to set the design framework let 119866 denote theglobal coordinate frame and let 119861 denote a coordinateframe attached to the vehicle usually denominated as body-fixed coordinate frameThe frame 119874 is the transceiver coor-dinates The position of transponder in the global coordinateframe is given by

119866119883119879=119866119883119863minus

119866

119861119877119861119883119863+119866

119861119877(

119861

119874119877119874119883119879+119861119883119874) (25)

where 119866119883119863is the position of the GPS in global coordinates

119861119883119863is the position of the GPS in vehicle coordinates 119874119883

119879

is the position of the transponder in transceiver coordinates119861119883119874is the position of the transceiver in vehicle coordinates

119866

119861119877is the rotation matrix from 119861 to 119866 and 119861

119874119877is the

rotation matrix from 119874 to 119861After the data collection we can get the distance between

the transponder and transceiver and the corresponding GPSposition in the global coordinates The attitude of the vehiclecan be measured by the heading sensors and the pitchrollsensor so 119866

119861119877is known We also know 119861119883

119863 119861119883119874 and 119861

119874119877

According to (25) we have

119874119883119879=

119861

119874119877minus1(

119866

119861119877minus1(119866119883119879minus119866119883119863) +119861119883119863minus119861119883119874) (26)

We denote 119874119883119879119894= [119874119909119879119894

119874119910119879119894

119874119911119879119894]

119879

(119894 = 1 2 119873)

as the position of the transponder in the transceiver coor-dinates in the 119894th measurement point The cost functionbecomes

119865 =

119873

sum

119894=1

(radic119874

1199092

119879119894+

119874

1199102

119879119894+

119874

1199112

119879119894minus 119871119894)

2

119873

(27)

Particles optimization algorithms have been introducedDefining each particle as a coordinate of the transponderTheparameters of particle swarmoptimization for simulationare the same as in Section 4 And a surface ship moves ina circular motion with a radius of 100 the real coordinatesof transponder are 119866119883

119903119879= [0 0 100]

119879 To simplify theproblem 119866

119861119877is unit matrix and 119861119883

119863minus119861119883119874= [0 0 0]

119879and ignore the sensor measurement error The simulationdata is shown in Table 3 Obviously the particle swarmalgorithm can search for the transponder coordinates andobtain accurate results At the same time CPSO shows thebest performance

Remark 2 One has

1198901 =119866119909119879minus119866119909119903119879 1198902 =

119866119910119879minus119866119910119903119879

1198903 =119866119911119879minus119866119911119903119879 119904119902 = radic1198901

2+ 11989022+ 11989032

(28)

7 Conclusion

In this paper a novel strategy to improve the performance ofparticle swarm optimization is proposed to apply in calibra-tion of the underwater transponder coordinates To improvethe population-based search optimization algorithm eachparticle is evolved along two different directions to generatetwo homologous particles The cost functions of two homol-ogous particles are calculated to keep the optimal one andto eliminate the poor one Then the next generation particleis updated It is regarded as CPSO Ten classify benchmarkfunctions are introduced to reflect the effectiveness of theproposed algorithmThe simulation results demonstrate thatthe unimodal function CPSO algorithm is superior toBPSO LWPSO EPSO and TVAC on the searching accuracystability and convergence speed However considering themultimodal function the performance of TVAC is superiorto CPSO

Mathematical Problems in Engineering 11

Secondly to further improve the performance of CPSOthe ECPSO is proposed by combining the CPSO and theTVAC In the initial period of the evolution the individualexperience is a significant aspect with larger accelerationcoefficient and in the final period the swarm experience issuperior with a greater acceleration coefficient Simultane-ously the evolution for each particle at any time is towardstwo different inertia directions to generate two homologousparticles and to obtain its next generation particles Thesimulations show the effectiveness of multimodal function byusing ECPSOWith the incensement of benchmark functionsdimensions the accuracy and stability of each algorithm willdecrease but CPSO and EPSO display the best performance

At last the strategy to calibrate the underwater transpon-der coordinates using particle swarm algorithm is intro-duced As the cost function for transponder coordinates is aunimodal function CPSO shows better performance than theother algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the Natural Science Foun-dation of China (51179038 1109043 51309067E091002) andthe Program of New Century Excellent Talents in University(NCET-10-0053)

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the 1995 IEEE International Conference on NeuralNetworks vol 4 no 2 pp 1942ndash1948 December 1995

[2] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[3] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[4] M S Arumugam M V C Rao and A W C Tan ldquoA noveland effective particle swarm optimization like algorithm withextrapolation techniquerdquo Applied Soft Computing Journal vol9 no 1 pp 308ndash320 2009

[5] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 1999

[6] G Chen X Huang J Jia et al ldquoNatural exponential inertiaweight strategy in particle swarm optimizationrdquo in Proceedingsof the 6thWorld Congress on Intelligent Control and Automation(WCICA rsquo06) vol 1 pp 3672ndash3675 IEEE June 2006

[7] Q Ni and J Deng ldquoA new logistic dynamic particle swarm opti-mization algorithm based on random topologyrdquo The ScientificWorld Journal vol 2013 Article ID 409167 8 pages 2013

[8] M M Noel ldquoA new gradient based particle swarm optimiza-tion algorithm for accurate computation of global minimumrdquoApplied Soft Computing Journal vol 12 no 1 pp 353ndash359 2012

[9] M G Epitropakis V P Plagianakos and M N VrahatisldquoEvolving cognitive and social experience in particle swarmoptimization through differential evolutionrdquo in Proceedings ofthe IEEE Congress on Evolutionary Computation (CEC rsquo10) pp1ndash8 IEEE July 2010

[10] A A Mousa M A El-Shorbagy and W F Abd-El-WahedldquoLocal search based hybrid particle swarm optimization algo-rithm for multiobjective optimizationrdquo Swarm and Evolution-ary Computation vol 3 pp 1ndash14 2012

[11] M H Moradi and M Abedini ldquoA combination of genetic algo-rithm and particle swarm optimization for optimal DG locationand sizing in distribution systemsrdquo International Journal ofElectrical Power and Energy Systems vol 34 no 1 pp 66ndash742012

[12] W D Chang J P Cheng M C Hsu et al ldquoParameteridentification of nonlinear systems using a particle swarmoptimization approachrdquo in Proceedings of the 3rd InternationalConference on Networking and Computing (ICNC rsquo12) pp 113ndash117 IEEE 2012

[13] J J Soon and K S Low ldquoPhotovoltaic model identificationusing particle swarm optimization with inverse barrier con-straintrdquo IEEE Transactions on Power Electronics vol 27 no 9pp 3975ndash3983 2012

[14] B Jiang N Wang and L Wang ldquoParameter identification forsolid oxide fuel cells using cooperative barebone particle swarmoptimization with hybrid learningrdquo International Journal ofHydrogen Energy vol 39 no 1 pp 532ndash542 2013

[15] A Alfi ldquoParticle swarm optimization algorithm with DynamicInertia Weight for online parameter identification applied toLorenz chaotic systemrdquo International Journal of InnovativeComputing Information and Control vol 8 no 2 pp 1191ndash12042012

[16] X Hu S Shi and X Gu ldquoAn improved particle swarm opti-mization algorithm for wireless sensor networks localizationrdquoin Proceedings of the 8th International Conference on WirelessCommunications Networking and Mobile Computing (WiCOMrsquo12) pp 1ndash4 Shanghai China September 2012

[17] F Sun J Yu and D Xu ldquoVisual measurement and control forunderwater robots a surveyrdquo in Proceedings of the 25th ChineseControl andDecision Conference (CCDC rsquo13) pp 333ndash338 IEEE2013

[18] V Djapic D Nad G Ferri et al ldquoNovel method for underwaternavigation aiding using a companion underwater robot asa guiding platformsrdquo in Proceedings of the 2013 MTSIEEEOCEANS-Bergen pp 1ndash10 IEEE June 2013

[19] A Caiti V Calabro D Meucci et al ldquoUnderwater Robots PastPresent and Futurerdquo 422ndash437

[20] P Krishnamurthy and F Khorrami ldquoA self-aligning underwaternavigation system based on fusion ofmultiple sensors includingDVL and IMUrdquo in Proceedings of the 9th Asian Control Confer-ence (ASCC rsquo13) pp 1ndash6 IEEE 2013

[21] S Wang and F Kang ldquoResearch and design of UUV navigationand control integrative simulation system based on compo-nentrdquo Intelligent Information Management vol 4 no 5 pp 181ndash187 2012

[22] T Kashima A Asada and T Ura ldquoThe positioning sys-tem integrated LBL and SSBL using seafloor acoustic mirrortransponderrdquo in Proceedings of the IEEE International Underwa-ter Technology Symposium (UT rsquo13) March 2013

12 Mathematical Problems in Engineering

[23] P Batista C Silvestre and P Oliveira ldquoGAS tightly coupledLBLUSBL position and velocity filter for underwater vehiclesrdquoin Proceedings of the European Control Conference (ECC rsquo13) pp2982ndash2987 Zurich Switzerland 2013

[24] M Morgado P Oliveira and C Silvestre ldquoTightly coupledultrashort baseline and inertial navigation system for under-water vehicles an experimental validationrdquo Journal of FieldRobotics vol 30 no 1 pp 142ndash170 2013

[25] Y U Min and H Junyin ldquoThe calibration of the USBL trans-ducer array for Long-range precision underwater positioningrdquoin Proceedings of the IEEE 10th International Conference onSignal Processing (ICSP rsquo10) pp 2357ndash2360 IEEE October 2010

[26] D Ji Y Li and J Liu ldquoSeafloor transponder calibration usingimproved perpendiculars intersectionrdquoAppliedOceanResearchvol 32 no 3 pp 261ndash266 2010

[27] B Jiao Z Lian and Q Chen ldquoA dynamic global and local com-bined particle swarm optimization algorithmrdquo Chaos Solitonsand Fractals vol 42 no 5 pp 2688ndash2695 2009

[28] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 IEEE May 1998

[29] M Schwaab E C Biscaia Jr J L Monteiro and J CPinto ldquoNonlinear parameter estimation through particle swarmoptimizationrdquo Chemical Engineering Science vol 63 no 6 pp1542ndash1552 2008

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII pp591ndash600 Springer Berlin Germany 1998

[31] A Chander A Chatterjee and P Siarry ldquoA new social andmomentum component adaptive PSO algorithm for imagesegmentationrdquo Expert Systems with Applications vol 38 no 5pp 4998ndash5004 2011

[32] M Morgado P Batista P Oliveira and C Silvestre ldquoPositionUSBLDVL sensor-based navigation filter in the presence ofunknown ocean currentsrdquoAutomatica vol 47 no 12 pp 2604ndash2614 2011

[33] B Jalving K Gade O K Hagen and K Vestgard ldquoA toolboxof aiding techniques for the HUGIN AUV integrated inertialnavigation systemrdquo in Proceedings of the OCEANS 2003 vol 2pp 1146ndash1153 IEEE September 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

the index of the best particle in the swarmTheposition vectorand the velocity vector of the 119894th particle in the119889-dimensionalsearch space can be represented as 119909

119894= (1199091198941 1199091198942 1199091198943 119909

119894119889)

and V119894= (V1198941 V1198942 V1198943 V

119894119889) respectively The best position

of each particle (119901best) is 119901119894= (1199011198941 1199011198942 1199011198943 119901

119894119889) and

the fitness particle found so far at generation 119896 (119892best) is119901119892= (1199011198921 1199011198922 119901

119892119889) In each generation each particle is

updated by the following two equations

V119894119889 (119896 + 1) = V

119894119889 (119896) + 1198881

times 1199031times (119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 1198882times 1199032times (119901119892119889 (

119896) minus 119909119894119889 (119896))

(1)

119909119894119889 (119896 + 1) = 119909119894119889 (

119896) + V119894119889 (119896 + 1) (2)

The parameters 1198881and 1198882are constants known as acceler-

ation coefficients 1199031and 1199032are random values in the range

from 0 to 1 and the value of 1199031and 1199032is not the same for

every iteration Kennedy and Eberhart [1] suggested settingeither of the acceleration coefficients at 2 in order tomake themean of both stochastic factors in (1) unity so that particleswould over fly only half the time of search The first equationshows that in PSO the search toward the optimum solutionis guided by the previous velocity the cognitive componentand the social component

Since the introduction of the particle swarm optimiza-tion numerous variations of the algorithm have been devel-oped in the literature Eberhart and Shi showed that PSOsearches for wide areas effectively but tends to lack localsearch precision They proposed in that work a solution byintroducing 120596 an inertia factor In this paper we name it asbasic particle swarm optimization (BPSO)

V119894119889 (119896 + 1) = 120596 times V

119894119889 (119896) + 1198881

times 1199031times (119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 1198882times 1199032times (119901119892119889 (

119896) minus 119909119894119889 (119896))

119909119894119889 (119896 + 1) = 119909119894119889 (

119896) + V119894119889 (119896 + 1)

(3)

22 Time-Varying Inertia Weight (TVIW) The role of theinertia weight 120596 is considered very important in PSO conver-gence behavior The inertia weight is applied to control theimpact of the previous history of velocities on the currentvelocity large inertia weight facilitates global explorationwhile small one tends to facilitate local exploration In orderto assure that the particles converge to the best point inthe course of the search Shi and Eberhart [28] have foundthat time-varying inertia weight (TVIW) has a significantimprovement in the performance of PSO and proposedlinear decreasing inertia weight PSO (LWPSO) with a lineardecreasing value of 120596 This modification can increase theexploration of the parameter space during the initial searchiterations and increase the exploitation of the parameter spaceduring the final steps of the search [29] The mathematicalrepresentation of inertia weight is given as follows

120596 = (1205961minus 1205962) times (

MAXITER minus 119896MAXITER

) + 1205962 (4)

where 1205961and 120596

2are the initial and final values of the inertia

weight respectively 119896 is the current iteration number and

MAXITER is the maximum number of allowable iterationsShi and Eberhart [5] indicate that the optimal solution can beimproved by varying the value of 120596 from 09 at the beginningof the search to 04 at the end of the search formost problems

Chen et al [6] proposed natural exponential (base 119890)inertia weight strategies named EPSO and expressed as

120596 = 1205962+ (1205961minus 1205962) times exp[minus( 119870

(MAXITER4))

2

] (5)

23 Time-Varying Acceleration Coefficient (TVAC) In PSOthe particle was updated due to the cognitive component andthe social component Therefore proper control of these twocomponents is very important to find the optimum solutionaccurately and efficiently Ratnaweera et al [2] introduced atime-varying acceleration coefficient (TVAC) which reducesthe cognitive component and increases the social componentby changing the acceleration coefficients 119888

1and 119888

2with

the time evolution The objective of this development is toenhance the global search in the early part of the optimizationand to encourage the particles to converge toward the globaloptima at the end of the search The TVAC is representedusing the following equations

1198881= (119888max minus 119888min)

119896

MAXITR+ 119888min

1198882= (119888min minus 119888max)

119896

MAXITR+ 119888max

(6)

where 119888min and 119888max are constants 119896 is the current iterationnumber and MAXITR is the maximum number of allowableiterations

Simulations were carried out with numerical bench-marks to find out the best ranges of values for 119888

1and 1198882

From the results it was observed that the best solutions weredetermined when changing 119888

1from 25 to 05 and changing 119888

2

from 05 to 25 over the full search range

3 Proposed New Developments

It is clarified from (1) that particlersquos new velocity is correlatedwith three terms the particlersquos previous velocity the valueof the cognitive component and the value of the socialcomponent Therefore proper control method with inertiaweight factor and acceleration coefficients is significant tofind the optimum solution accurately and efficiently

The inertia weight is utilized to adjust the influence of theprevious velocity on the current velocity and balance betweenglobal and local exploration abilities of the ldquoflying particlerdquo[30 31] A larger inertia weight implies stronger globalexploration ability advocating the particle to escape from alocal minimum A smaller inertia weight leads to strongerlocal exploration ability confining the particle searchingwithin a local range near its present position to guarantee theconvergence

Kennedy and Eberhart [1] indicated that a relatively highvalue of the cognitive component compared with the socialcomponent will result in excessive wandering of individuals

4 Mathematical Problems in Engineering

through the search space In contrast a relatively high value ofthe social component may lead particles to rush prematurelytoward a local optimum

Considering those concerns we propose a new strategyfor the PSO concept

31 Competition Particle Swarm Optimization (CPSO) Inthe process of particle evolution each particle is evolvedalong different directions with different inertia coefficientsand acceleration coefficients Two homologous particles aregenerated and the optimal one is kept through comparingcost functions of two homologous particles eliminating theinferior one Then the next generation particle is updatedfinally The evaluation function of each particle is describedas

V119905119894119889(119896 + 1) = 120596

119905V119894119889 (119896) + 119888

119905

1119903119905

1(119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 119888119905

2119903119905

2(119901119892119889 (

119896) minus 119909119894119889 (119896))

119909119905

119894119889(119896 + 1) = 119909

119905

119894119889(119896) + V119905

119894119889(119896 + 1)

(7)

And the final equations are shown as

V119894119889 (119896) = V119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

119909119894119889 (119896) = 119909

119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

(8)

where 119899 is the number of particles in the swarm 119872 is themaximum iteration frequency 119903119905

1 1199031199052are the randomnumbers

in the range of [0 1] 119909119894119889(119896) is the 119894th particle position in the

119889th dimension after 119896 time iteration 119909119905119894119889(119896) shows the 119889th

dimension position for subparticle 119905 of the 119894th particle after119896 time iteration 120596119905 is the speed inertia weight of subparticle119905 1198881199051and 1198881199052are constants denoting acceleration coefficients

fitness (119909119905119894(119870)) is the fitting function of the subparticle 119905 after

119896 time iteration V119905119894119889(119896) is the speed of subparticle 119905 at 119889th

dimension 119901119894119889(119896) is the optimal position of the 119894th particle

at 119889th dimension and 119901119892119889(119896) is the swarm optimal position

at 119889th dimension after 119896 time iteration

Remark 1 In this paper two subparticles are generated foreach particle at one time therefore 119905 = 1 2

The detailed steps are shown as follows

Step 1 Initial

Substep 1 Set the initial parameters 119899119872 119908119905 1198881199051 1198881199052

Substep 2 Take random initial 119909119894119889(0)

Substep 3 Take random initial V119905119894119889(1)

Substep 4 Calculate fitness(119909119905119894(0)) and set 119901

119894(0) = 119909

119894(0)

Substep 5 One has 119901119892(0) = 119909

119894(0) | 119909

119894(0) isin

min(fitness(119909119894(0)))

Step 2 If the criteria are satisfied output the best solutionotherwise go to Substep 6

Substep 6 Update V119905119894119889(119896) and 119909119905

119894119889(119896)

Substep 7 Calculate fitness(119909119905119894(119896))

Substep 8 One has 119909119894119889(119896) = 119909

119905

119894119889(119896) | min(fitness(119909119905

119894(119896))) 119905 =

1 2

Substep 9 If fitness(119909119894(119896)) lt fitness(119901

119894(119896))

119901119894 (119896) = 119909119894 (

119896) (9)

If minfitness(119909119894(119896)) 119894 = 1 119899 lt fitness(119901

119892(119896))

119901119892 (119896) = 119909119894 (

119896) | min [fitness (119909119894 (119896)) 119894 = 1 119899] (10)

Substep 10 Go back to Step 2

32 Extension Competition Particle Swarm Optimization(ECPSO) competition particle swarm optimization (CPSO)helps adjust the search direction particles and improve thesearch speed and efficiency but due to rapid convergenceCPSO is easy to fall into local minima According tobenchmark functions simulation in Section 4 it is obviousthat CPSO is superior to the searching effective of single-modulus function and TVAC is superior to the searchingeffective of multimodulus function The reason resulting inthis phenomenon is due to the selection of accelerationcoefficient With a large cognitive component and a smallsocial component at the beginning particles are allowed tomove around the search space instead of moving towardthe population best On the other hand a small cognitivecomponent and a large social component allow the particlesto converge to the global optima in the latter part of theoptimization Considering the advantage of TVAC introduceTVAC into CPSO and the extension competition particleswarm optimization (ECPSO) with the acceleration coeffi-cients proposed as above The evolution equations can bemathematically represented as the following

V119905119894119889(119896 + 1) = 120596

119905V119894119889 (119896) + 1198881

119903119905

1(119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 1198882119903119905

2(119901119892119889 (

119896) minus 119909119894119889 (119896))

119909119905

119894119889(119896 + 1) = 119909

119905

119894119889(119896) + V119905

119894119889(119896 + 1)

(11)

where

1198881= (119888max minus 119888min)

119896

MAXITR+ 119888min

1198882= (119888min minus 119888max)

119896

MAXITR+ 119888max

(12)

And it is obvious that

V119894119889 (119896) = V119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

119909119894119889 (119896) = 119909

119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

(13)

Mathematical Problems in Engineering 5

4 Experimental Settings and SimulationStrategies for Benchmark Testing

Simulations were carried out to observe the rate of con-vergence and the quality of the optimum solution of thenew methods introduced in this investigation by comparingwith BPSO EPSO and TVAC From the standard set ofbenchmark problems available in the literature there are 5important functions considered to test the efficacy of theproposed method All of the benchmark functions reflectdifferent degrees of complexity

41 Functions Introduction The functions are as follows

(1) Sphere function one has

1198911 (119909) =

119863

sum

119894=1

1199092

119894 (14)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is verysimple convex and unimodal function with only one localoptimum value

(2) Axis parallel hyperellipsoid function one has

1198912 (119909) =

119863

sum

119894=1

119894 sdot 1199092

119894 (15)

It is known as the weighted sphere model With the searchspace 119909

119894| minus100 lt 119909

119894lt 100 the global minimum locates at

119909 = [0 0]119863 with 119891(119909) = 0 It is continuous convex and

unimodal

(3) Rotated hyperellipsoid function (Schwefelrsquos problem12) one has

1198913 (119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

(16)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is

continuous convex and unimodal With respect to the coor-dinate axes this function produces rotated hyperellipsoids

(4) Moved axis parallel hyperellipsoid function one has

1198914 (119909) =

119863

sum

119894=1

5119894 sdot 1199092

119894 (17)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909(119894) = 5 lowast 119894 119894 = 1 119863 with 119891(119909) = 0

(5) Rosenbrock function one has

1198915 (119909) =

119863minus1

sum

119894=1

[100(119909119894+1minus 1199092

119894)

2

+ (119909119894minus 1)2] (18)

Table 1 Parameters for simulation

Method Parameters

CPSO1205961= 120596max = 09 120596

2= 120596min = 04

1198881

1= 05 1198881

2= 25 1198882

1= 25

1198882

2= 05

119899 = 30119863 = 10

119909119894isin [minus100 100]

V119894isin [minus100 100]

ECPSO1205961= 120596max = 09 120596

2= 120596min = 04

119888min = 05 119888max = 25

BPSO 1198881= 1198882= 20 120596 = 07

EPSO1198881= 1198882= 20 120596max = 09120596min = 04

TVAC 119888min = 05 119888max = 25 120596 = 07

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [1 1]119863 with 119891(119909) = 0 It

is a unimodal function and the global optimum is inside along narrow parabolic shaped flat valley To find the valley istrivial

(6) Rastrigin function one has

1198916 (119909) =

119863

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894) + 10] (19)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is

highly multimodal However the locations of the minima areregularly distributed

(7) Griewank function one has

1198917 (119909) =

1

4000

119863

sum

119894=1

1199092

119894minus

119863

prod

119894=1

cos(119909119894

radic119894

) + 1 (20)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It

is a multimodal function and has many widespread localminima However the locations of the minima are regularlydistributed

(8) Sum of different power function one has

1198918 (119909) =

119863

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816

(119894+1) (21)

The sum of different powers is a commonly used uni-modal test function With the search space 119909

119894| minus100 lt

119909119894lt 100 the global minimum locates at 119909 = [0 0]

119863

with 119891(119909) = 0

(9) Ackleyrsquos path function one has

1198919 (119909) = minus119886 sdot 119890

minus119887sdotradicsum119863

119894=11199092

119894119863minus 119890sum119863

119894=1cos(119888sdot119909119894)119863

+ 119886 + 1198901

(22)

6 Mathematical Problems in Engineering

Table 2 Comparison of BPSO WPSO EPSO TVAC and CPSO

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198911

10 47979119890 minus 11

(11343119890 minus 10)32219119890 minus 24

(10706119890 minus 23)87747119890 minus 51

(3408119890 minus 50)14826119890 minus 54

(14770119890 minus 53)41023119890 minus 146

(36388119890 minus 145)41221119890 minus 118

(30916119890 minus 117)

30 53768(43459)

40137119890 minus 05

(5926119890 minus 05)23219119890 minus 12

(11495119890 minus 11)11062119890 minus 10

(73302119890 minus 10)14872119890 minus 46

(64922119890 minus 46)16829119890 minus 10

(10243119890 minus 09)

50 28412119890 + 01

(45047)02149(01634)

22215119890 minus 05

(35696119890 minus 05)00019(00079)

93999119890 minus 20

(40558119890 minus 19)00022(00083)

70 44686119890 + 01

(43454)11728119890 + 01

(12113119890 + 01)00192(00238)

00609(01033)

20133119890 minus 07

(19424119890 minus 06)00302(00458)

1198912

10 37589119890 minus 10

(17479119890 minus 09)85629119890 minus 24

(19031119890 minus 23)52910119890 minus 50

(23037119890 minus 49)15947119890 minus 52

(14618119890 minus 51)89632119890 minus 146

(35734119890 minus 145)30640119890 minus 113

(30622119890 minus 112)

30 378269(313869)

00005(00007)

10278119890 minus 11

(20979119890 minus 11)11937119890 minus 09

(65345119890 minus 09)81119119890 minus 45

(44882119890 minus 44)30047119890 minus 10

(21285119890 minus 09)

50 68495119890 + 02

(10535119890 + 02)28041(22863)

00004(00010)

01214(06006)

24666119890 minus 15

(24614119890 minus 14)00223(00835)

70 14952119890 + 03

(18073119890 + 02)12961119890 + 02

(14505119890 + 02)02601(02405)

21146(39189)

12547119890 minus 05

(00001)11879(20240)

1198913

10 00013(00018)

40624119890 minus 08

(11066119890 minus 07)31542119890 minus 15

(19686119890 minus 14)36072119890 minus 27

(16376119890 minus 26)60846119890 minus 60

(58552119890 minus 59)44092119890 minus 51

(31687119890 minus 50)

30 20647119890 + 01

(79831)37596(20255)

05809(03373)

00091(00197)

27957119890 minus 06

(50455119890 minus 06)00258(00385)

50 72983119890 + 01

(28263119890 + 01)26189119890 + 01

(13125119890 + 01)10295119890 + 01

(51265)04883(03021)

00775(00620)

05720(03509)

70 14821119890 + 02

(46282119890 + 01)69337119890 + 01

(33718119890 + 01)29336119890 + 01

(10417119890 + 01)27039(10853)

11214(04917)

22342(07423)

1198914

10 27409119890 minus 09

(15090119890 minus 08)13708119890 minus 22

(65937119890 minus 22)35012119890 minus 49

(28064119890 minus 48)99156119890 minus 52

(69209119890 minus 51)16140119890 minus 144

(14296119890 minus 143)30561119890 minus 115

(21383119890 minus 114)

30 19825119890 + 02

(17202119890 + 02)00025(00029)

73319119890 minus 11

(23165119890 minus 10)66962119890 minus 09

(36744119890 minus 08)10828119890 minus 42

(89052119890 minus 42)56960119890 minus 08

(39998119890 minus 07)

50 32922119890 + 03

(56799119890 + 02)13397119890 + 01

(94756)00015(00019)

03064(12614)

22477119890 minus 17

(14523119890 minus 16)02379(11928)

70 74756119890 + 03

(75131119890 + 02)68348119890 + 02

(61444119890 + 02)16799(29749)

10304119890 + 01

(14839119890 + 01)19407119890 minus 06

(96092119890 minus 06)55072

(10286119890 + 01)

1198915

10 56418(1092)

43037(12401)

31701(12637)

07262(09490)

06503(14786)

06445(14706)

30 12322119890 + 03

(94694119890 + 02)42313119890 + 01

(26596119890 + 01)31052119890 + 01

(17699119890 + 01)23370119890 + 01

(19238)14702119890 + 01

(31506)19234119890 + 01

(29573)

50 65575119890 + 03

(13919119890 + 03)29055119890 + 02

(16229119890 + 02)73894119890 + 01

(36633119890 + 01)46563119890 + 01

(25861)374782119890 + 01

(27904)47741119890 + 01

(106763)

70 11834119890 + 04

(19184119890 + 03)52635119890 + 03

(40246119890 + 03)20414119890 + 02

(74873119890 + 01)75240119890 + 01

(12063119890 + 01)65838119890 + 01

(15847119890 + 01)84068119890 + 01

(30477119890 + 01)

1198916

10 51090(30667)

36806(17685)

38703(17882)

09750(09693)

22685(13272)

06765(08231)

30 15696119890 + 02

(39880119890 + 01)32291119890 + 01

(10059119890 + 01)29949119890 + 01

(71988)17282119890 + 01

(54892)18148119890 + 01

(62546)13312119890 + 01

(44942)

50 38006119890 + 02

(35258119890 + 01)75141119890 + 01

(2587119890 + 01)61896119890 + 01

(15107119890 + 01)36575119890 + 01

(94137)36236119890 + 01

(91136)30739119890 + 01

(85709)

70 58512119890 + 02

(36592119890 + 01)16259119890 + 02

(62498119890 + 01)88994119890 + 01

(17552119890 + 01)54198119890 + 01

(11482119890 + 01)52752119890 + 01

(12361119890 + 01)49043119890 + 01

(11607119890 + 01)

1198917

10 98646119890 minus 05

(00009)00041(00096)

00017(00055)

0(0)

0(0)

0(0)

30 01366(01090)

00008(00045)

00013(00071)

10210119890 minus 10

(10021119890 minus 09)35527119890 minus 17

(14708119890 minus 16)17067119890 minus 11

(13918119890 minus 10)

50 05856(00891)

00027(00021)

99034119890 minus 05

(00009)00001(00007)

24547119890 minus 15

(88945119890 minus 15)36456119890 minus 05

(00001)

70 06941(00646)

00329(00190)

00002(00010)

00011(00017)

44137119890 minus 11

(27956119890 minus 10)00007(00012)

Mathematical Problems in Engineering 7

Table 2 Continued

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198918

10 58516119890 minus 18

(25897119890 minus 17)22299119890 minus 40

(16366119890 minus 39)20832119890 minus 84

(16585119890 minus 83)25229119890 minus 87

(11045119890 minus 86)58546119890 minus 237

(0)22665119890 minus 154

(22664119890 minus 153)

30 19895119890 + 02

(62708119890 + 02)10418119890 minus 06

(37845119890 minus 06)73163119890 minus 18

(27837119890 minus 17)10483119890 minus 39

(72123119890 minus 39)30968119890 minus 99

(30467119890 minus 98)10029119890 minus 41

(10029119890 minus 40)

50 19927119890 + 07

(71603119890 + 07)18348119890 + 01

(48704119890 + 01)00017(00067)

65668119890 minus 25

(31648119890 minus 24)15985119890 minus 47

(14843119890 minus 46)37443119890 minus 23

(37143119890 minus 22)

70 15637119890 + 13

(83656119890 + 13)26886119890 + 08

(16837119890 + 09)14973119890 + 04

(10807119890 + 05)49509119890 minus 19

(26306119890 minus 18)32094119890 minus 29

(25130119890 minus 28)19384119890 minus 19

(10414119890 minus 18)

1198919

10 00472(01618)

01115(02369)

01773(02354)

50448119890 minus 15

(13412119890 minus 15)01269(02045)

00132(00770)

30 15006(02835)

05798(00888)

05469(00765)

03709(00736)

04187(00747)

03822(00767)

50 18625(01067)

06039(01209)

05348(00655)

03691(00544)

03848(00541)

03631(00478)

70 18765(00728)

08228(02349)

04790(00512)

03439(00463)

03369(00469)

03393(00413)

11989110

10 94158119890 minus 13

(49298119890 minus 12)14531119890 minus 26

(39844119890 minus 26)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)25554119890 minus 33

(10557119890 minus 32)

30 00527(00668)

34402119890 minus 07

(54202119890 minus 07)99564119890 minus 15

(20677119890 minus 14)67912119890 minus 12

(46051119890 minus 11)33310119890 minus 31

(14574119890 minus 30)83251119890 minus 13

(42490119890 minus 12)

50 13060(04239)

00010(00008)

18317119890 minus 07

(34350119890 minus 07)00003(00015)

00018(00127)

00011(00091)

70 26961(04601)

00350(00393)

00011(00091)

00059(00115)

00009(00090)

00023(00094)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0Set the coefficients as 119886 = 20 119887 = 02 119888 = 2 sdot 120587

(10) Penalised function one has

11989110 (119909) = 01sin2 (31205871199091) +

119863minus1

sum

119894=1

(119909119894minus 1)2[1 + sin2 (3120587119909

1)]

+ (119909119863minus 1)2[1 + sin2 (2120587119909

119863)]

+

119863

sum

119894=1

119906 (119909119894 5 100 4)

(23)

where

119906 (119909119894 119886 119896 119898) =

119896(119909119894minus 119886)119898 119909

119894gt 119886

0 minus119886 le 119909119894le 119886

119896(minus119909119894minus 119886)119898 119909119894lt minus119886

119910119894= 1 +

1

4

(119909119894+ 1)

(24)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [1 1]119863 with 119891(119909) = 0

42 The Coefficients Setting The parameters for simulationare listed in Table 1

In this table 119888(sdot)

expresses the accelerations coefficients120596 denotes inertia weight the dimension is 119863 and the rangeof the search space and the velocity space are 119909

(sdot)and V(sdot) If

the current position is out of the search space the positionof the particle is taken to be the value of the boundaryand the velocity is taken to be zero If the velocity of theparticle is outside of the boundary its value is set to bethe boundary value The maximum number of iterations isset to 1000 For each function 100 trials were carried outand the average optimal value and the standard deviationare presented To verify the performance of the algorithm atdifferent dimensions variable dimension 119863 increases from10 to 100 and the optimal mean and variance of benchmarkfunctions are calculated The results are presented in Table 2

5 The Results in Comparison withthe Previous Developments

The simulation results are given in Table 2 The comparisonresults elucidate that the searching accuracy and stabilityranging from low to high are listed as BPSO LWPSO EPSOTVAC ECPSO and CPSO for unimodal function It isobvious that the performances of ECPSO and CPSO aresuperior due to their advantage of obtaining the optimalspeed direction and the searching efficiency while in themultimodal function the CPSO algorithm is easy to trap intolocal minimum and TVAC shows better performance thanCPSO Combining the advantages of CPSO and TVAC theEPSO algorithm is applied to enhance the global search in

8 Mathematical Problems in Engineering

0 200 400 600 800 1000

0

05

1

15

2

25

3

35

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

0 200 400 600 800 1000

0

2

4

6

8

10

12

14

16

18

Number of generations

Mea

n be

st va

lue

0 200 400 600 800 1000

0

1

2

3

4

5

Number of generations

Number of generations Number of generations

Number of generations

0 200 400 600 800 1000

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000

0

100

200

300

400

500

0 200 400 600 800 1000

0

10

20

30

40

50

60

minus05

minus10

minus10minus100

minus1

minus2

f1

f5

f4f3

f2

f6

Figure 1 Continued

Mathematical Problems in Engineering 9

0 200 400 600 800 1000

0

005

01

015

02

025

03

035

Number of generations

Number of generations Number of generations

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

0 200 400 600 800 1000

0

05

1

15

2

25

3

0 200 400 600 800 1000

0

05

1

15

2

25

0 200 400 600 800 1000

0

002

004

006

008

01

012

014

016

minus05

minus05

minus005

minus002

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

f8f7

f10f9

Figure 1 Variation of the average optimum value with time

the early part of the optimization and encourage the particlesto converge toward the global optima at the end of the searchComparing to TVAC andCPSO the proposed new algorithmECPSO is appropriate for multimodal function search

With the increase of benchmark functions dimensionthe searching accuracy and stability for each algorithmare decreased The performances of CPSO and ECPSO aresuperior to the other algorithms With regard to multimodalfunction the performances of ECPSOandTVACare superiorto CPSO The average fitness is varied as in Table 2 FromFigure 1 it shows that the order of searching speed fromhigh to low is BPSO (green solid curve) LWPSO (red circle)EPSO (blue triangle) TVAC (black dot curve) CPSO (bluedash-dot) and ECPSO (purples dash) It is obvious that theperformances of EPSO and CPSO are more effective than theother algorithms

6 Calibrate the UnderwaterTransponder Coordinates

After two decades of dedicated research and developmentunmanned underwater vehicles (UUV) have been acceptedby an increasing number of users in bothmilitary and civilianinstitutions The design and implementation of navigationsystems stand out as one of the most critical steps towardsthe successful operation of autonomous vehicles The qualityof the overall estimates of the navigation system dramaticallyinfluences the capability of the vehicles to perform precision-demanding tasks [32] From a navigation point of view asingle range transponder may be regarded as an underwaterlighthouse providingUUVwith the ranges relative to its fixedgeographical location Single transponder navigation is not anew concept As mentioned the first at-sea demonstration of

10 Mathematical Problems in Engineering

Table 3 The calibration of the underwater transponder coordinates

BPSO LWPSO EPSO TVAC CPSO ECPSO1198901 35371119890 minus 15 minus23573119890 minus 15 26542119890 minus 15 minus19813119890 minus 15 minus11700119890 minus 15 14526119890 minus 15

1198902 15510119890 minus 15 23280119890 minus 15 minus48804119890 minus 16 minus14289119890 minus 15 13702119890 minus 15 22549119890 minus 15

1198903 0 0 0 0 0 0119904119902 41948119890 minus 15 27057119890 minus 15 26986119890 minus 15 24428119890 minus 15 18018119890 minus 15 26823119890 minus 15

Layingpoint

z

yG

x

Surfaceship

Transponder

Seabed

Figure 2 The calibration geometry for calibration of the transpon-der position

single transponder UTP aided inertial navigation was carriedout in 2003 as described in [33]

For ranging techniques like UTP to work the geo-graphical location of the transponders must be known Thepreferred method is to measure the position directly usingUSBL on a surface ship When the transponder deploymentis completed a surface ship with USBL and GPS sails aroundthe transponder in a circular motion collecting surface shipposition and distance information The calibration geometryis illustrated in Figure 2

In order to set the design framework let 119866 denote theglobal coordinate frame and let 119861 denote a coordinateframe attached to the vehicle usually denominated as body-fixed coordinate frameThe frame 119874 is the transceiver coor-dinates The position of transponder in the global coordinateframe is given by

119866119883119879=119866119883119863minus

119866

119861119877119861119883119863+119866

119861119877(

119861

119874119877119874119883119879+119861119883119874) (25)

where 119866119883119863is the position of the GPS in global coordinates

119861119883119863is the position of the GPS in vehicle coordinates 119874119883

119879

is the position of the transponder in transceiver coordinates119861119883119874is the position of the transceiver in vehicle coordinates

119866

119861119877is the rotation matrix from 119861 to 119866 and 119861

119874119877is the

rotation matrix from 119874 to 119861After the data collection we can get the distance between

the transponder and transceiver and the corresponding GPSposition in the global coordinates The attitude of the vehiclecan be measured by the heading sensors and the pitchrollsensor so 119866

119861119877is known We also know 119861119883

119863 119861119883119874 and 119861

119874119877

According to (25) we have

119874119883119879=

119861

119874119877minus1(

119866

119861119877minus1(119866119883119879minus119866119883119863) +119861119883119863minus119861119883119874) (26)

We denote 119874119883119879119894= [119874119909119879119894

119874119910119879119894

119874119911119879119894]

119879

(119894 = 1 2 119873)

as the position of the transponder in the transceiver coor-dinates in the 119894th measurement point The cost functionbecomes

119865 =

119873

sum

119894=1

(radic119874

1199092

119879119894+

119874

1199102

119879119894+

119874

1199112

119879119894minus 119871119894)

2

119873

(27)

Particles optimization algorithms have been introducedDefining each particle as a coordinate of the transponderTheparameters of particle swarmoptimization for simulationare the same as in Section 4 And a surface ship moves ina circular motion with a radius of 100 the real coordinatesof transponder are 119866119883

119903119879= [0 0 100]

119879 To simplify theproblem 119866

119861119877is unit matrix and 119861119883

119863minus119861119883119874= [0 0 0]

119879and ignore the sensor measurement error The simulationdata is shown in Table 3 Obviously the particle swarmalgorithm can search for the transponder coordinates andobtain accurate results At the same time CPSO shows thebest performance

Remark 2 One has

1198901 =119866119909119879minus119866119909119903119879 1198902 =

119866119910119879minus119866119910119903119879

1198903 =119866119911119879minus119866119911119903119879 119904119902 = radic1198901

2+ 11989022+ 11989032

(28)

7 Conclusion

In this paper a novel strategy to improve the performance ofparticle swarm optimization is proposed to apply in calibra-tion of the underwater transponder coordinates To improvethe population-based search optimization algorithm eachparticle is evolved along two different directions to generatetwo homologous particles The cost functions of two homol-ogous particles are calculated to keep the optimal one andto eliminate the poor one Then the next generation particleis updated It is regarded as CPSO Ten classify benchmarkfunctions are introduced to reflect the effectiveness of theproposed algorithmThe simulation results demonstrate thatthe unimodal function CPSO algorithm is superior toBPSO LWPSO EPSO and TVAC on the searching accuracystability and convergence speed However considering themultimodal function the performance of TVAC is superiorto CPSO

Mathematical Problems in Engineering 11

Secondly to further improve the performance of CPSOthe ECPSO is proposed by combining the CPSO and theTVAC In the initial period of the evolution the individualexperience is a significant aspect with larger accelerationcoefficient and in the final period the swarm experience issuperior with a greater acceleration coefficient Simultane-ously the evolution for each particle at any time is towardstwo different inertia directions to generate two homologousparticles and to obtain its next generation particles Thesimulations show the effectiveness of multimodal function byusing ECPSOWith the incensement of benchmark functionsdimensions the accuracy and stability of each algorithm willdecrease but CPSO and EPSO display the best performance

At last the strategy to calibrate the underwater transpon-der coordinates using particle swarm algorithm is intro-duced As the cost function for transponder coordinates is aunimodal function CPSO shows better performance than theother algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the Natural Science Foun-dation of China (51179038 1109043 51309067E091002) andthe Program of New Century Excellent Talents in University(NCET-10-0053)

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the 1995 IEEE International Conference on NeuralNetworks vol 4 no 2 pp 1942ndash1948 December 1995

[2] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[3] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[4] M S Arumugam M V C Rao and A W C Tan ldquoA noveland effective particle swarm optimization like algorithm withextrapolation techniquerdquo Applied Soft Computing Journal vol9 no 1 pp 308ndash320 2009

[5] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 1999

[6] G Chen X Huang J Jia et al ldquoNatural exponential inertiaweight strategy in particle swarm optimizationrdquo in Proceedingsof the 6thWorld Congress on Intelligent Control and Automation(WCICA rsquo06) vol 1 pp 3672ndash3675 IEEE June 2006

[7] Q Ni and J Deng ldquoA new logistic dynamic particle swarm opti-mization algorithm based on random topologyrdquo The ScientificWorld Journal vol 2013 Article ID 409167 8 pages 2013

[8] M M Noel ldquoA new gradient based particle swarm optimiza-tion algorithm for accurate computation of global minimumrdquoApplied Soft Computing Journal vol 12 no 1 pp 353ndash359 2012

[9] M G Epitropakis V P Plagianakos and M N VrahatisldquoEvolving cognitive and social experience in particle swarmoptimization through differential evolutionrdquo in Proceedings ofthe IEEE Congress on Evolutionary Computation (CEC rsquo10) pp1ndash8 IEEE July 2010

[10] A A Mousa M A El-Shorbagy and W F Abd-El-WahedldquoLocal search based hybrid particle swarm optimization algo-rithm for multiobjective optimizationrdquo Swarm and Evolution-ary Computation vol 3 pp 1ndash14 2012

[11] M H Moradi and M Abedini ldquoA combination of genetic algo-rithm and particle swarm optimization for optimal DG locationand sizing in distribution systemsrdquo International Journal ofElectrical Power and Energy Systems vol 34 no 1 pp 66ndash742012

[12] W D Chang J P Cheng M C Hsu et al ldquoParameteridentification of nonlinear systems using a particle swarmoptimization approachrdquo in Proceedings of the 3rd InternationalConference on Networking and Computing (ICNC rsquo12) pp 113ndash117 IEEE 2012

[13] J J Soon and K S Low ldquoPhotovoltaic model identificationusing particle swarm optimization with inverse barrier con-straintrdquo IEEE Transactions on Power Electronics vol 27 no 9pp 3975ndash3983 2012

[14] B Jiang N Wang and L Wang ldquoParameter identification forsolid oxide fuel cells using cooperative barebone particle swarmoptimization with hybrid learningrdquo International Journal ofHydrogen Energy vol 39 no 1 pp 532ndash542 2013

[15] A Alfi ldquoParticle swarm optimization algorithm with DynamicInertia Weight for online parameter identification applied toLorenz chaotic systemrdquo International Journal of InnovativeComputing Information and Control vol 8 no 2 pp 1191ndash12042012

[16] X Hu S Shi and X Gu ldquoAn improved particle swarm opti-mization algorithm for wireless sensor networks localizationrdquoin Proceedings of the 8th International Conference on WirelessCommunications Networking and Mobile Computing (WiCOMrsquo12) pp 1ndash4 Shanghai China September 2012

[17] F Sun J Yu and D Xu ldquoVisual measurement and control forunderwater robots a surveyrdquo in Proceedings of the 25th ChineseControl andDecision Conference (CCDC rsquo13) pp 333ndash338 IEEE2013

[18] V Djapic D Nad G Ferri et al ldquoNovel method for underwaternavigation aiding using a companion underwater robot asa guiding platformsrdquo in Proceedings of the 2013 MTSIEEEOCEANS-Bergen pp 1ndash10 IEEE June 2013

[19] A Caiti V Calabro D Meucci et al ldquoUnderwater Robots PastPresent and Futurerdquo 422ndash437

[20] P Krishnamurthy and F Khorrami ldquoA self-aligning underwaternavigation system based on fusion ofmultiple sensors includingDVL and IMUrdquo in Proceedings of the 9th Asian Control Confer-ence (ASCC rsquo13) pp 1ndash6 IEEE 2013

[21] S Wang and F Kang ldquoResearch and design of UUV navigationand control integrative simulation system based on compo-nentrdquo Intelligent Information Management vol 4 no 5 pp 181ndash187 2012

[22] T Kashima A Asada and T Ura ldquoThe positioning sys-tem integrated LBL and SSBL using seafloor acoustic mirrortransponderrdquo in Proceedings of the IEEE International Underwa-ter Technology Symposium (UT rsquo13) March 2013

12 Mathematical Problems in Engineering

[23] P Batista C Silvestre and P Oliveira ldquoGAS tightly coupledLBLUSBL position and velocity filter for underwater vehiclesrdquoin Proceedings of the European Control Conference (ECC rsquo13) pp2982ndash2987 Zurich Switzerland 2013

[24] M Morgado P Oliveira and C Silvestre ldquoTightly coupledultrashort baseline and inertial navigation system for under-water vehicles an experimental validationrdquo Journal of FieldRobotics vol 30 no 1 pp 142ndash170 2013

[25] Y U Min and H Junyin ldquoThe calibration of the USBL trans-ducer array for Long-range precision underwater positioningrdquoin Proceedings of the IEEE 10th International Conference onSignal Processing (ICSP rsquo10) pp 2357ndash2360 IEEE October 2010

[26] D Ji Y Li and J Liu ldquoSeafloor transponder calibration usingimproved perpendiculars intersectionrdquoAppliedOceanResearchvol 32 no 3 pp 261ndash266 2010

[27] B Jiao Z Lian and Q Chen ldquoA dynamic global and local com-bined particle swarm optimization algorithmrdquo Chaos Solitonsand Fractals vol 42 no 5 pp 2688ndash2695 2009

[28] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 IEEE May 1998

[29] M Schwaab E C Biscaia Jr J L Monteiro and J CPinto ldquoNonlinear parameter estimation through particle swarmoptimizationrdquo Chemical Engineering Science vol 63 no 6 pp1542ndash1552 2008

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII pp591ndash600 Springer Berlin Germany 1998

[31] A Chander A Chatterjee and P Siarry ldquoA new social andmomentum component adaptive PSO algorithm for imagesegmentationrdquo Expert Systems with Applications vol 38 no 5pp 4998ndash5004 2011

[32] M Morgado P Batista P Oliveira and C Silvestre ldquoPositionUSBLDVL sensor-based navigation filter in the presence ofunknown ocean currentsrdquoAutomatica vol 47 no 12 pp 2604ndash2614 2011

[33] B Jalving K Gade O K Hagen and K Vestgard ldquoA toolboxof aiding techniques for the HUGIN AUV integrated inertialnavigation systemrdquo in Proceedings of the OCEANS 2003 vol 2pp 1146ndash1153 IEEE September 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

through the search space In contrast a relatively high value ofthe social component may lead particles to rush prematurelytoward a local optimum

Considering those concerns we propose a new strategyfor the PSO concept

31 Competition Particle Swarm Optimization (CPSO) Inthe process of particle evolution each particle is evolvedalong different directions with different inertia coefficientsand acceleration coefficients Two homologous particles aregenerated and the optimal one is kept through comparingcost functions of two homologous particles eliminating theinferior one Then the next generation particle is updatedfinally The evaluation function of each particle is describedas

V119905119894119889(119896 + 1) = 120596

119905V119894119889 (119896) + 119888

119905

1119903119905

1(119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 119888119905

2119903119905

2(119901119892119889 (

119896) minus 119909119894119889 (119896))

119909119905

119894119889(119896 + 1) = 119909

119905

119894119889(119896) + V119905

119894119889(119896 + 1)

(7)

And the final equations are shown as

V119894119889 (119896) = V119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

119909119894119889 (119896) = 119909

119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

(8)

where 119899 is the number of particles in the swarm 119872 is themaximum iteration frequency 119903119905

1 1199031199052are the randomnumbers

in the range of [0 1] 119909119894119889(119896) is the 119894th particle position in the

119889th dimension after 119896 time iteration 119909119905119894119889(119896) shows the 119889th

dimension position for subparticle 119905 of the 119894th particle after119896 time iteration 120596119905 is the speed inertia weight of subparticle119905 1198881199051and 1198881199052are constants denoting acceleration coefficients

fitness (119909119905119894(119870)) is the fitting function of the subparticle 119905 after

119896 time iteration V119905119894119889(119896) is the speed of subparticle 119905 at 119889th

dimension 119901119894119889(119896) is the optimal position of the 119894th particle

at 119889th dimension and 119901119892119889(119896) is the swarm optimal position

at 119889th dimension after 119896 time iteration

Remark 1 In this paper two subparticles are generated foreach particle at one time therefore 119905 = 1 2

The detailed steps are shown as follows

Step 1 Initial

Substep 1 Set the initial parameters 119899119872 119908119905 1198881199051 1198881199052

Substep 2 Take random initial 119909119894119889(0)

Substep 3 Take random initial V119905119894119889(1)

Substep 4 Calculate fitness(119909119905119894(0)) and set 119901

119894(0) = 119909

119894(0)

Substep 5 One has 119901119892(0) = 119909

119894(0) | 119909

119894(0) isin

min(fitness(119909119894(0)))

Step 2 If the criteria are satisfied output the best solutionotherwise go to Substep 6

Substep 6 Update V119905119894119889(119896) and 119909119905

119894119889(119896)

Substep 7 Calculate fitness(119909119905119894(119896))

Substep 8 One has 119909119894119889(119896) = 119909

119905

119894119889(119896) | min(fitness(119909119905

119894(119896))) 119905 =

1 2

Substep 9 If fitness(119909119894(119896)) lt fitness(119901

119894(119896))

119901119894 (119896) = 119909119894 (

119896) (9)

If minfitness(119909119894(119896)) 119894 = 1 119899 lt fitness(119901

119892(119896))

119901119892 (119896) = 119909119894 (

119896) | min [fitness (119909119894 (119896)) 119894 = 1 119899] (10)

Substep 10 Go back to Step 2

32 Extension Competition Particle Swarm Optimization(ECPSO) competition particle swarm optimization (CPSO)helps adjust the search direction particles and improve thesearch speed and efficiency but due to rapid convergenceCPSO is easy to fall into local minima According tobenchmark functions simulation in Section 4 it is obviousthat CPSO is superior to the searching effective of single-modulus function and TVAC is superior to the searchingeffective of multimodulus function The reason resulting inthis phenomenon is due to the selection of accelerationcoefficient With a large cognitive component and a smallsocial component at the beginning particles are allowed tomove around the search space instead of moving towardthe population best On the other hand a small cognitivecomponent and a large social component allow the particlesto converge to the global optima in the latter part of theoptimization Considering the advantage of TVAC introduceTVAC into CPSO and the extension competition particleswarm optimization (ECPSO) with the acceleration coeffi-cients proposed as above The evolution equations can bemathematically represented as the following

V119905119894119889(119896 + 1) = 120596

119905V119894119889 (119896) + 1198881

119903119905

1(119901119894119889 (119896) minus 119909119894119889 (

119896))

+ 1198882119903119905

2(119901119892119889 (

119896) minus 119909119894119889 (119896))

119909119905

119894119889(119896 + 1) = 119909

119905

119894119889(119896) + V119905

119894119889(119896 + 1)

(11)

where

1198881= (119888max minus 119888min)

119896

MAXITR+ 119888min

1198882= (119888min minus 119888max)

119896

MAXITR+ 119888max

(12)

And it is obvious that

V119894119889 (119896) = V119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

119909119894119889 (119896) = 119909

119905

119894119889(119896) | min (fitness (119909119905

119894(119870))) 119905 = 1 2

(13)

Mathematical Problems in Engineering 5

4 Experimental Settings and SimulationStrategies for Benchmark Testing

Simulations were carried out to observe the rate of con-vergence and the quality of the optimum solution of thenew methods introduced in this investigation by comparingwith BPSO EPSO and TVAC From the standard set ofbenchmark problems available in the literature there are 5important functions considered to test the efficacy of theproposed method All of the benchmark functions reflectdifferent degrees of complexity

41 Functions Introduction The functions are as follows

(1) Sphere function one has

1198911 (119909) =

119863

sum

119894=1

1199092

119894 (14)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is verysimple convex and unimodal function with only one localoptimum value

(2) Axis parallel hyperellipsoid function one has

1198912 (119909) =

119863

sum

119894=1

119894 sdot 1199092

119894 (15)

It is known as the weighted sphere model With the searchspace 119909

119894| minus100 lt 119909

119894lt 100 the global minimum locates at

119909 = [0 0]119863 with 119891(119909) = 0 It is continuous convex and

unimodal

(3) Rotated hyperellipsoid function (Schwefelrsquos problem12) one has

1198913 (119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

(16)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is

continuous convex and unimodal With respect to the coor-dinate axes this function produces rotated hyperellipsoids

(4) Moved axis parallel hyperellipsoid function one has

1198914 (119909) =

119863

sum

119894=1

5119894 sdot 1199092

119894 (17)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909(119894) = 5 lowast 119894 119894 = 1 119863 with 119891(119909) = 0

(5) Rosenbrock function one has

1198915 (119909) =

119863minus1

sum

119894=1

[100(119909119894+1minus 1199092

119894)

2

+ (119909119894minus 1)2] (18)

Table 1 Parameters for simulation

Method Parameters

CPSO1205961= 120596max = 09 120596

2= 120596min = 04

1198881

1= 05 1198881

2= 25 1198882

1= 25

1198882

2= 05

119899 = 30119863 = 10

119909119894isin [minus100 100]

V119894isin [minus100 100]

ECPSO1205961= 120596max = 09 120596

2= 120596min = 04

119888min = 05 119888max = 25

BPSO 1198881= 1198882= 20 120596 = 07

EPSO1198881= 1198882= 20 120596max = 09120596min = 04

TVAC 119888min = 05 119888max = 25 120596 = 07

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [1 1]119863 with 119891(119909) = 0 It

is a unimodal function and the global optimum is inside along narrow parabolic shaped flat valley To find the valley istrivial

(6) Rastrigin function one has

1198916 (119909) =

119863

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894) + 10] (19)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is

highly multimodal However the locations of the minima areregularly distributed

(7) Griewank function one has

1198917 (119909) =

1

4000

119863

sum

119894=1

1199092

119894minus

119863

prod

119894=1

cos(119909119894

radic119894

) + 1 (20)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It

is a multimodal function and has many widespread localminima However the locations of the minima are regularlydistributed

(8) Sum of different power function one has

1198918 (119909) =

119863

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816

(119894+1) (21)

The sum of different powers is a commonly used uni-modal test function With the search space 119909

119894| minus100 lt

119909119894lt 100 the global minimum locates at 119909 = [0 0]

119863

with 119891(119909) = 0

(9) Ackleyrsquos path function one has

1198919 (119909) = minus119886 sdot 119890

minus119887sdotradicsum119863

119894=11199092

119894119863minus 119890sum119863

119894=1cos(119888sdot119909119894)119863

+ 119886 + 1198901

(22)

6 Mathematical Problems in Engineering

Table 2 Comparison of BPSO WPSO EPSO TVAC and CPSO

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198911

10 47979119890 minus 11

(11343119890 minus 10)32219119890 minus 24

(10706119890 minus 23)87747119890 minus 51

(3408119890 minus 50)14826119890 minus 54

(14770119890 minus 53)41023119890 minus 146

(36388119890 minus 145)41221119890 minus 118

(30916119890 minus 117)

30 53768(43459)

40137119890 minus 05

(5926119890 minus 05)23219119890 minus 12

(11495119890 minus 11)11062119890 minus 10

(73302119890 minus 10)14872119890 minus 46

(64922119890 minus 46)16829119890 minus 10

(10243119890 minus 09)

50 28412119890 + 01

(45047)02149(01634)

22215119890 minus 05

(35696119890 minus 05)00019(00079)

93999119890 minus 20

(40558119890 minus 19)00022(00083)

70 44686119890 + 01

(43454)11728119890 + 01

(12113119890 + 01)00192(00238)

00609(01033)

20133119890 minus 07

(19424119890 minus 06)00302(00458)

1198912

10 37589119890 minus 10

(17479119890 minus 09)85629119890 minus 24

(19031119890 minus 23)52910119890 minus 50

(23037119890 minus 49)15947119890 minus 52

(14618119890 minus 51)89632119890 minus 146

(35734119890 minus 145)30640119890 minus 113

(30622119890 minus 112)

30 378269(313869)

00005(00007)

10278119890 minus 11

(20979119890 minus 11)11937119890 minus 09

(65345119890 minus 09)81119119890 minus 45

(44882119890 minus 44)30047119890 minus 10

(21285119890 minus 09)

50 68495119890 + 02

(10535119890 + 02)28041(22863)

00004(00010)

01214(06006)

24666119890 minus 15

(24614119890 minus 14)00223(00835)

70 14952119890 + 03

(18073119890 + 02)12961119890 + 02

(14505119890 + 02)02601(02405)

21146(39189)

12547119890 minus 05

(00001)11879(20240)

1198913

10 00013(00018)

40624119890 minus 08

(11066119890 minus 07)31542119890 minus 15

(19686119890 minus 14)36072119890 minus 27

(16376119890 minus 26)60846119890 minus 60

(58552119890 minus 59)44092119890 minus 51

(31687119890 minus 50)

30 20647119890 + 01

(79831)37596(20255)

05809(03373)

00091(00197)

27957119890 minus 06

(50455119890 minus 06)00258(00385)

50 72983119890 + 01

(28263119890 + 01)26189119890 + 01

(13125119890 + 01)10295119890 + 01

(51265)04883(03021)

00775(00620)

05720(03509)

70 14821119890 + 02

(46282119890 + 01)69337119890 + 01

(33718119890 + 01)29336119890 + 01

(10417119890 + 01)27039(10853)

11214(04917)

22342(07423)

1198914

10 27409119890 minus 09

(15090119890 minus 08)13708119890 minus 22

(65937119890 minus 22)35012119890 minus 49

(28064119890 minus 48)99156119890 minus 52

(69209119890 minus 51)16140119890 minus 144

(14296119890 minus 143)30561119890 minus 115

(21383119890 minus 114)

30 19825119890 + 02

(17202119890 + 02)00025(00029)

73319119890 minus 11

(23165119890 minus 10)66962119890 minus 09

(36744119890 minus 08)10828119890 minus 42

(89052119890 minus 42)56960119890 minus 08

(39998119890 minus 07)

50 32922119890 + 03

(56799119890 + 02)13397119890 + 01

(94756)00015(00019)

03064(12614)

22477119890 minus 17

(14523119890 minus 16)02379(11928)

70 74756119890 + 03

(75131119890 + 02)68348119890 + 02

(61444119890 + 02)16799(29749)

10304119890 + 01

(14839119890 + 01)19407119890 minus 06

(96092119890 minus 06)55072

(10286119890 + 01)

1198915

10 56418(1092)

43037(12401)

31701(12637)

07262(09490)

06503(14786)

06445(14706)

30 12322119890 + 03

(94694119890 + 02)42313119890 + 01

(26596119890 + 01)31052119890 + 01

(17699119890 + 01)23370119890 + 01

(19238)14702119890 + 01

(31506)19234119890 + 01

(29573)

50 65575119890 + 03

(13919119890 + 03)29055119890 + 02

(16229119890 + 02)73894119890 + 01

(36633119890 + 01)46563119890 + 01

(25861)374782119890 + 01

(27904)47741119890 + 01

(106763)

70 11834119890 + 04

(19184119890 + 03)52635119890 + 03

(40246119890 + 03)20414119890 + 02

(74873119890 + 01)75240119890 + 01

(12063119890 + 01)65838119890 + 01

(15847119890 + 01)84068119890 + 01

(30477119890 + 01)

1198916

10 51090(30667)

36806(17685)

38703(17882)

09750(09693)

22685(13272)

06765(08231)

30 15696119890 + 02

(39880119890 + 01)32291119890 + 01

(10059119890 + 01)29949119890 + 01

(71988)17282119890 + 01

(54892)18148119890 + 01

(62546)13312119890 + 01

(44942)

50 38006119890 + 02

(35258119890 + 01)75141119890 + 01

(2587119890 + 01)61896119890 + 01

(15107119890 + 01)36575119890 + 01

(94137)36236119890 + 01

(91136)30739119890 + 01

(85709)

70 58512119890 + 02

(36592119890 + 01)16259119890 + 02

(62498119890 + 01)88994119890 + 01

(17552119890 + 01)54198119890 + 01

(11482119890 + 01)52752119890 + 01

(12361119890 + 01)49043119890 + 01

(11607119890 + 01)

1198917

10 98646119890 minus 05

(00009)00041(00096)

00017(00055)

0(0)

0(0)

0(0)

30 01366(01090)

00008(00045)

00013(00071)

10210119890 minus 10

(10021119890 minus 09)35527119890 minus 17

(14708119890 minus 16)17067119890 minus 11

(13918119890 minus 10)

50 05856(00891)

00027(00021)

99034119890 minus 05

(00009)00001(00007)

24547119890 minus 15

(88945119890 minus 15)36456119890 minus 05

(00001)

70 06941(00646)

00329(00190)

00002(00010)

00011(00017)

44137119890 minus 11

(27956119890 minus 10)00007(00012)

Mathematical Problems in Engineering 7

Table 2 Continued

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198918

10 58516119890 minus 18

(25897119890 minus 17)22299119890 minus 40

(16366119890 minus 39)20832119890 minus 84

(16585119890 minus 83)25229119890 minus 87

(11045119890 minus 86)58546119890 minus 237

(0)22665119890 minus 154

(22664119890 minus 153)

30 19895119890 + 02

(62708119890 + 02)10418119890 minus 06

(37845119890 minus 06)73163119890 minus 18

(27837119890 minus 17)10483119890 minus 39

(72123119890 minus 39)30968119890 minus 99

(30467119890 minus 98)10029119890 minus 41

(10029119890 minus 40)

50 19927119890 + 07

(71603119890 + 07)18348119890 + 01

(48704119890 + 01)00017(00067)

65668119890 minus 25

(31648119890 minus 24)15985119890 minus 47

(14843119890 minus 46)37443119890 minus 23

(37143119890 minus 22)

70 15637119890 + 13

(83656119890 + 13)26886119890 + 08

(16837119890 + 09)14973119890 + 04

(10807119890 + 05)49509119890 minus 19

(26306119890 minus 18)32094119890 minus 29

(25130119890 minus 28)19384119890 minus 19

(10414119890 minus 18)

1198919

10 00472(01618)

01115(02369)

01773(02354)

50448119890 minus 15

(13412119890 minus 15)01269(02045)

00132(00770)

30 15006(02835)

05798(00888)

05469(00765)

03709(00736)

04187(00747)

03822(00767)

50 18625(01067)

06039(01209)

05348(00655)

03691(00544)

03848(00541)

03631(00478)

70 18765(00728)

08228(02349)

04790(00512)

03439(00463)

03369(00469)

03393(00413)

11989110

10 94158119890 minus 13

(49298119890 minus 12)14531119890 minus 26

(39844119890 minus 26)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)25554119890 minus 33

(10557119890 minus 32)

30 00527(00668)

34402119890 minus 07

(54202119890 minus 07)99564119890 minus 15

(20677119890 minus 14)67912119890 minus 12

(46051119890 minus 11)33310119890 minus 31

(14574119890 minus 30)83251119890 minus 13

(42490119890 minus 12)

50 13060(04239)

00010(00008)

18317119890 minus 07

(34350119890 minus 07)00003(00015)

00018(00127)

00011(00091)

70 26961(04601)

00350(00393)

00011(00091)

00059(00115)

00009(00090)

00023(00094)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0Set the coefficients as 119886 = 20 119887 = 02 119888 = 2 sdot 120587

(10) Penalised function one has

11989110 (119909) = 01sin2 (31205871199091) +

119863minus1

sum

119894=1

(119909119894minus 1)2[1 + sin2 (3120587119909

1)]

+ (119909119863minus 1)2[1 + sin2 (2120587119909

119863)]

+

119863

sum

119894=1

119906 (119909119894 5 100 4)

(23)

where

119906 (119909119894 119886 119896 119898) =

119896(119909119894minus 119886)119898 119909

119894gt 119886

0 minus119886 le 119909119894le 119886

119896(minus119909119894minus 119886)119898 119909119894lt minus119886

119910119894= 1 +

1

4

(119909119894+ 1)

(24)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [1 1]119863 with 119891(119909) = 0

42 The Coefficients Setting The parameters for simulationare listed in Table 1

In this table 119888(sdot)

expresses the accelerations coefficients120596 denotes inertia weight the dimension is 119863 and the rangeof the search space and the velocity space are 119909

(sdot)and V(sdot) If

the current position is out of the search space the positionof the particle is taken to be the value of the boundaryand the velocity is taken to be zero If the velocity of theparticle is outside of the boundary its value is set to bethe boundary value The maximum number of iterations isset to 1000 For each function 100 trials were carried outand the average optimal value and the standard deviationare presented To verify the performance of the algorithm atdifferent dimensions variable dimension 119863 increases from10 to 100 and the optimal mean and variance of benchmarkfunctions are calculated The results are presented in Table 2

5 The Results in Comparison withthe Previous Developments

The simulation results are given in Table 2 The comparisonresults elucidate that the searching accuracy and stabilityranging from low to high are listed as BPSO LWPSO EPSOTVAC ECPSO and CPSO for unimodal function It isobvious that the performances of ECPSO and CPSO aresuperior due to their advantage of obtaining the optimalspeed direction and the searching efficiency while in themultimodal function the CPSO algorithm is easy to trap intolocal minimum and TVAC shows better performance thanCPSO Combining the advantages of CPSO and TVAC theEPSO algorithm is applied to enhance the global search in

8 Mathematical Problems in Engineering

0 200 400 600 800 1000

0

05

1

15

2

25

3

35

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

0 200 400 600 800 1000

0

2

4

6

8

10

12

14

16

18

Number of generations

Mea

n be

st va

lue

0 200 400 600 800 1000

0

1

2

3

4

5

Number of generations

Number of generations Number of generations

Number of generations

0 200 400 600 800 1000

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000

0

100

200

300

400

500

0 200 400 600 800 1000

0

10

20

30

40

50

60

minus05

minus10

minus10minus100

minus1

minus2

f1

f5

f4f3

f2

f6

Figure 1 Continued

Mathematical Problems in Engineering 9

0 200 400 600 800 1000

0

005

01

015

02

025

03

035

Number of generations

Number of generations Number of generations

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

0 200 400 600 800 1000

0

05

1

15

2

25

3

0 200 400 600 800 1000

0

05

1

15

2

25

0 200 400 600 800 1000

0

002

004

006

008

01

012

014

016

minus05

minus05

minus005

minus002

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

f8f7

f10f9

Figure 1 Variation of the average optimum value with time

the early part of the optimization and encourage the particlesto converge toward the global optima at the end of the searchComparing to TVAC andCPSO the proposed new algorithmECPSO is appropriate for multimodal function search

With the increase of benchmark functions dimensionthe searching accuracy and stability for each algorithmare decreased The performances of CPSO and ECPSO aresuperior to the other algorithms With regard to multimodalfunction the performances of ECPSOandTVACare superiorto CPSO The average fitness is varied as in Table 2 FromFigure 1 it shows that the order of searching speed fromhigh to low is BPSO (green solid curve) LWPSO (red circle)EPSO (blue triangle) TVAC (black dot curve) CPSO (bluedash-dot) and ECPSO (purples dash) It is obvious that theperformances of EPSO and CPSO are more effective than theother algorithms

6 Calibrate the UnderwaterTransponder Coordinates

After two decades of dedicated research and developmentunmanned underwater vehicles (UUV) have been acceptedby an increasing number of users in bothmilitary and civilianinstitutions The design and implementation of navigationsystems stand out as one of the most critical steps towardsthe successful operation of autonomous vehicles The qualityof the overall estimates of the navigation system dramaticallyinfluences the capability of the vehicles to perform precision-demanding tasks [32] From a navigation point of view asingle range transponder may be regarded as an underwaterlighthouse providingUUVwith the ranges relative to its fixedgeographical location Single transponder navigation is not anew concept As mentioned the first at-sea demonstration of

10 Mathematical Problems in Engineering

Table 3 The calibration of the underwater transponder coordinates

BPSO LWPSO EPSO TVAC CPSO ECPSO1198901 35371119890 minus 15 minus23573119890 minus 15 26542119890 minus 15 minus19813119890 minus 15 minus11700119890 minus 15 14526119890 minus 15

1198902 15510119890 minus 15 23280119890 minus 15 minus48804119890 minus 16 minus14289119890 minus 15 13702119890 minus 15 22549119890 minus 15

1198903 0 0 0 0 0 0119904119902 41948119890 minus 15 27057119890 minus 15 26986119890 minus 15 24428119890 minus 15 18018119890 minus 15 26823119890 minus 15

Layingpoint

z

yG

x

Surfaceship

Transponder

Seabed

Figure 2 The calibration geometry for calibration of the transpon-der position

single transponder UTP aided inertial navigation was carriedout in 2003 as described in [33]

For ranging techniques like UTP to work the geo-graphical location of the transponders must be known Thepreferred method is to measure the position directly usingUSBL on a surface ship When the transponder deploymentis completed a surface ship with USBL and GPS sails aroundthe transponder in a circular motion collecting surface shipposition and distance information The calibration geometryis illustrated in Figure 2

In order to set the design framework let 119866 denote theglobal coordinate frame and let 119861 denote a coordinateframe attached to the vehicle usually denominated as body-fixed coordinate frameThe frame 119874 is the transceiver coor-dinates The position of transponder in the global coordinateframe is given by

119866119883119879=119866119883119863minus

119866

119861119877119861119883119863+119866

119861119877(

119861

119874119877119874119883119879+119861119883119874) (25)

where 119866119883119863is the position of the GPS in global coordinates

119861119883119863is the position of the GPS in vehicle coordinates 119874119883

119879

is the position of the transponder in transceiver coordinates119861119883119874is the position of the transceiver in vehicle coordinates

119866

119861119877is the rotation matrix from 119861 to 119866 and 119861

119874119877is the

rotation matrix from 119874 to 119861After the data collection we can get the distance between

the transponder and transceiver and the corresponding GPSposition in the global coordinates The attitude of the vehiclecan be measured by the heading sensors and the pitchrollsensor so 119866

119861119877is known We also know 119861119883

119863 119861119883119874 and 119861

119874119877

According to (25) we have

119874119883119879=

119861

119874119877minus1(

119866

119861119877minus1(119866119883119879minus119866119883119863) +119861119883119863minus119861119883119874) (26)

We denote 119874119883119879119894= [119874119909119879119894

119874119910119879119894

119874119911119879119894]

119879

(119894 = 1 2 119873)

as the position of the transponder in the transceiver coor-dinates in the 119894th measurement point The cost functionbecomes

119865 =

119873

sum

119894=1

(radic119874

1199092

119879119894+

119874

1199102

119879119894+

119874

1199112

119879119894minus 119871119894)

2

119873

(27)

Particles optimization algorithms have been introducedDefining each particle as a coordinate of the transponderTheparameters of particle swarmoptimization for simulationare the same as in Section 4 And a surface ship moves ina circular motion with a radius of 100 the real coordinatesof transponder are 119866119883

119903119879= [0 0 100]

119879 To simplify theproblem 119866

119861119877is unit matrix and 119861119883

119863minus119861119883119874= [0 0 0]

119879and ignore the sensor measurement error The simulationdata is shown in Table 3 Obviously the particle swarmalgorithm can search for the transponder coordinates andobtain accurate results At the same time CPSO shows thebest performance

Remark 2 One has

1198901 =119866119909119879minus119866119909119903119879 1198902 =

119866119910119879minus119866119910119903119879

1198903 =119866119911119879minus119866119911119903119879 119904119902 = radic1198901

2+ 11989022+ 11989032

(28)

7 Conclusion

In this paper a novel strategy to improve the performance ofparticle swarm optimization is proposed to apply in calibra-tion of the underwater transponder coordinates To improvethe population-based search optimization algorithm eachparticle is evolved along two different directions to generatetwo homologous particles The cost functions of two homol-ogous particles are calculated to keep the optimal one andto eliminate the poor one Then the next generation particleis updated It is regarded as CPSO Ten classify benchmarkfunctions are introduced to reflect the effectiveness of theproposed algorithmThe simulation results demonstrate thatthe unimodal function CPSO algorithm is superior toBPSO LWPSO EPSO and TVAC on the searching accuracystability and convergence speed However considering themultimodal function the performance of TVAC is superiorto CPSO

Mathematical Problems in Engineering 11

Secondly to further improve the performance of CPSOthe ECPSO is proposed by combining the CPSO and theTVAC In the initial period of the evolution the individualexperience is a significant aspect with larger accelerationcoefficient and in the final period the swarm experience issuperior with a greater acceleration coefficient Simultane-ously the evolution for each particle at any time is towardstwo different inertia directions to generate two homologousparticles and to obtain its next generation particles Thesimulations show the effectiveness of multimodal function byusing ECPSOWith the incensement of benchmark functionsdimensions the accuracy and stability of each algorithm willdecrease but CPSO and EPSO display the best performance

At last the strategy to calibrate the underwater transpon-der coordinates using particle swarm algorithm is intro-duced As the cost function for transponder coordinates is aunimodal function CPSO shows better performance than theother algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the Natural Science Foun-dation of China (51179038 1109043 51309067E091002) andthe Program of New Century Excellent Talents in University(NCET-10-0053)

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the 1995 IEEE International Conference on NeuralNetworks vol 4 no 2 pp 1942ndash1948 December 1995

[2] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[3] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[4] M S Arumugam M V C Rao and A W C Tan ldquoA noveland effective particle swarm optimization like algorithm withextrapolation techniquerdquo Applied Soft Computing Journal vol9 no 1 pp 308ndash320 2009

[5] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 1999

[6] G Chen X Huang J Jia et al ldquoNatural exponential inertiaweight strategy in particle swarm optimizationrdquo in Proceedingsof the 6thWorld Congress on Intelligent Control and Automation(WCICA rsquo06) vol 1 pp 3672ndash3675 IEEE June 2006

[7] Q Ni and J Deng ldquoA new logistic dynamic particle swarm opti-mization algorithm based on random topologyrdquo The ScientificWorld Journal vol 2013 Article ID 409167 8 pages 2013

[8] M M Noel ldquoA new gradient based particle swarm optimiza-tion algorithm for accurate computation of global minimumrdquoApplied Soft Computing Journal vol 12 no 1 pp 353ndash359 2012

[9] M G Epitropakis V P Plagianakos and M N VrahatisldquoEvolving cognitive and social experience in particle swarmoptimization through differential evolutionrdquo in Proceedings ofthe IEEE Congress on Evolutionary Computation (CEC rsquo10) pp1ndash8 IEEE July 2010

[10] A A Mousa M A El-Shorbagy and W F Abd-El-WahedldquoLocal search based hybrid particle swarm optimization algo-rithm for multiobjective optimizationrdquo Swarm and Evolution-ary Computation vol 3 pp 1ndash14 2012

[11] M H Moradi and M Abedini ldquoA combination of genetic algo-rithm and particle swarm optimization for optimal DG locationand sizing in distribution systemsrdquo International Journal ofElectrical Power and Energy Systems vol 34 no 1 pp 66ndash742012

[12] W D Chang J P Cheng M C Hsu et al ldquoParameteridentification of nonlinear systems using a particle swarmoptimization approachrdquo in Proceedings of the 3rd InternationalConference on Networking and Computing (ICNC rsquo12) pp 113ndash117 IEEE 2012

[13] J J Soon and K S Low ldquoPhotovoltaic model identificationusing particle swarm optimization with inverse barrier con-straintrdquo IEEE Transactions on Power Electronics vol 27 no 9pp 3975ndash3983 2012

[14] B Jiang N Wang and L Wang ldquoParameter identification forsolid oxide fuel cells using cooperative barebone particle swarmoptimization with hybrid learningrdquo International Journal ofHydrogen Energy vol 39 no 1 pp 532ndash542 2013

[15] A Alfi ldquoParticle swarm optimization algorithm with DynamicInertia Weight for online parameter identification applied toLorenz chaotic systemrdquo International Journal of InnovativeComputing Information and Control vol 8 no 2 pp 1191ndash12042012

[16] X Hu S Shi and X Gu ldquoAn improved particle swarm opti-mization algorithm for wireless sensor networks localizationrdquoin Proceedings of the 8th International Conference on WirelessCommunications Networking and Mobile Computing (WiCOMrsquo12) pp 1ndash4 Shanghai China September 2012

[17] F Sun J Yu and D Xu ldquoVisual measurement and control forunderwater robots a surveyrdquo in Proceedings of the 25th ChineseControl andDecision Conference (CCDC rsquo13) pp 333ndash338 IEEE2013

[18] V Djapic D Nad G Ferri et al ldquoNovel method for underwaternavigation aiding using a companion underwater robot asa guiding platformsrdquo in Proceedings of the 2013 MTSIEEEOCEANS-Bergen pp 1ndash10 IEEE June 2013

[19] A Caiti V Calabro D Meucci et al ldquoUnderwater Robots PastPresent and Futurerdquo 422ndash437

[20] P Krishnamurthy and F Khorrami ldquoA self-aligning underwaternavigation system based on fusion ofmultiple sensors includingDVL and IMUrdquo in Proceedings of the 9th Asian Control Confer-ence (ASCC rsquo13) pp 1ndash6 IEEE 2013

[21] S Wang and F Kang ldquoResearch and design of UUV navigationand control integrative simulation system based on compo-nentrdquo Intelligent Information Management vol 4 no 5 pp 181ndash187 2012

[22] T Kashima A Asada and T Ura ldquoThe positioning sys-tem integrated LBL and SSBL using seafloor acoustic mirrortransponderrdquo in Proceedings of the IEEE International Underwa-ter Technology Symposium (UT rsquo13) March 2013

12 Mathematical Problems in Engineering

[23] P Batista C Silvestre and P Oliveira ldquoGAS tightly coupledLBLUSBL position and velocity filter for underwater vehiclesrdquoin Proceedings of the European Control Conference (ECC rsquo13) pp2982ndash2987 Zurich Switzerland 2013

[24] M Morgado P Oliveira and C Silvestre ldquoTightly coupledultrashort baseline and inertial navigation system for under-water vehicles an experimental validationrdquo Journal of FieldRobotics vol 30 no 1 pp 142ndash170 2013

[25] Y U Min and H Junyin ldquoThe calibration of the USBL trans-ducer array for Long-range precision underwater positioningrdquoin Proceedings of the IEEE 10th International Conference onSignal Processing (ICSP rsquo10) pp 2357ndash2360 IEEE October 2010

[26] D Ji Y Li and J Liu ldquoSeafloor transponder calibration usingimproved perpendiculars intersectionrdquoAppliedOceanResearchvol 32 no 3 pp 261ndash266 2010

[27] B Jiao Z Lian and Q Chen ldquoA dynamic global and local com-bined particle swarm optimization algorithmrdquo Chaos Solitonsand Fractals vol 42 no 5 pp 2688ndash2695 2009

[28] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 IEEE May 1998

[29] M Schwaab E C Biscaia Jr J L Monteiro and J CPinto ldquoNonlinear parameter estimation through particle swarmoptimizationrdquo Chemical Engineering Science vol 63 no 6 pp1542ndash1552 2008

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII pp591ndash600 Springer Berlin Germany 1998

[31] A Chander A Chatterjee and P Siarry ldquoA new social andmomentum component adaptive PSO algorithm for imagesegmentationrdquo Expert Systems with Applications vol 38 no 5pp 4998ndash5004 2011

[32] M Morgado P Batista P Oliveira and C Silvestre ldquoPositionUSBLDVL sensor-based navigation filter in the presence ofunknown ocean currentsrdquoAutomatica vol 47 no 12 pp 2604ndash2614 2011

[33] B Jalving K Gade O K Hagen and K Vestgard ldquoA toolboxof aiding techniques for the HUGIN AUV integrated inertialnavigation systemrdquo in Proceedings of the OCEANS 2003 vol 2pp 1146ndash1153 IEEE September 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

4 Experimental Settings and SimulationStrategies for Benchmark Testing

Simulations were carried out to observe the rate of con-vergence and the quality of the optimum solution of thenew methods introduced in this investigation by comparingwith BPSO EPSO and TVAC From the standard set ofbenchmark problems available in the literature there are 5important functions considered to test the efficacy of theproposed method All of the benchmark functions reflectdifferent degrees of complexity

41 Functions Introduction The functions are as follows

(1) Sphere function one has

1198911 (119909) =

119863

sum

119894=1

1199092

119894 (14)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is verysimple convex and unimodal function with only one localoptimum value

(2) Axis parallel hyperellipsoid function one has

1198912 (119909) =

119863

sum

119894=1

119894 sdot 1199092

119894 (15)

It is known as the weighted sphere model With the searchspace 119909

119894| minus100 lt 119909

119894lt 100 the global minimum locates at

119909 = [0 0]119863 with 119891(119909) = 0 It is continuous convex and

unimodal

(3) Rotated hyperellipsoid function (Schwefelrsquos problem12) one has

1198913 (119909) =

119863

sum

119894=1

(

119894

sum

119895=1

119909119895)

2

(16)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is

continuous convex and unimodal With respect to the coor-dinate axes this function produces rotated hyperellipsoids

(4) Moved axis parallel hyperellipsoid function one has

1198914 (119909) =

119863

sum

119894=1

5119894 sdot 1199092

119894 (17)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909(119894) = 5 lowast 119894 119894 = 1 119863 with 119891(119909) = 0

(5) Rosenbrock function one has

1198915 (119909) =

119863minus1

sum

119894=1

[100(119909119894+1minus 1199092

119894)

2

+ (119909119894minus 1)2] (18)

Table 1 Parameters for simulation

Method Parameters

CPSO1205961= 120596max = 09 120596

2= 120596min = 04

1198881

1= 05 1198881

2= 25 1198882

1= 25

1198882

2= 05

119899 = 30119863 = 10

119909119894isin [minus100 100]

V119894isin [minus100 100]

ECPSO1205961= 120596max = 09 120596

2= 120596min = 04

119888min = 05 119888max = 25

BPSO 1198881= 1198882= 20 120596 = 07

EPSO1198881= 1198882= 20 120596max = 09120596min = 04

TVAC 119888min = 05 119888max = 25 120596 = 07

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [1 1]119863 with 119891(119909) = 0 It

is a unimodal function and the global optimum is inside along narrow parabolic shaped flat valley To find the valley istrivial

(6) Rastrigin function one has

1198916 (119909) =

119863

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894) + 10] (19)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It is

highly multimodal However the locations of the minima areregularly distributed

(7) Griewank function one has

1198917 (119909) =

1

4000

119863

sum

119894=1

1199092

119894minus

119863

prod

119894=1

cos(119909119894

radic119894

) + 1 (20)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0 It

is a multimodal function and has many widespread localminima However the locations of the minima are regularlydistributed

(8) Sum of different power function one has

1198918 (119909) =

119863

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816

(119894+1) (21)

The sum of different powers is a commonly used uni-modal test function With the search space 119909

119894| minus100 lt

119909119894lt 100 the global minimum locates at 119909 = [0 0]

119863

with 119891(119909) = 0

(9) Ackleyrsquos path function one has

1198919 (119909) = minus119886 sdot 119890

minus119887sdotradicsum119863

119894=11199092

119894119863minus 119890sum119863

119894=1cos(119888sdot119909119894)119863

+ 119886 + 1198901

(22)

6 Mathematical Problems in Engineering

Table 2 Comparison of BPSO WPSO EPSO TVAC and CPSO

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198911

10 47979119890 minus 11

(11343119890 minus 10)32219119890 minus 24

(10706119890 minus 23)87747119890 minus 51

(3408119890 minus 50)14826119890 minus 54

(14770119890 minus 53)41023119890 minus 146

(36388119890 minus 145)41221119890 minus 118

(30916119890 minus 117)

30 53768(43459)

40137119890 minus 05

(5926119890 minus 05)23219119890 minus 12

(11495119890 minus 11)11062119890 minus 10

(73302119890 minus 10)14872119890 minus 46

(64922119890 minus 46)16829119890 minus 10

(10243119890 minus 09)

50 28412119890 + 01

(45047)02149(01634)

22215119890 minus 05

(35696119890 minus 05)00019(00079)

93999119890 minus 20

(40558119890 minus 19)00022(00083)

70 44686119890 + 01

(43454)11728119890 + 01

(12113119890 + 01)00192(00238)

00609(01033)

20133119890 minus 07

(19424119890 minus 06)00302(00458)

1198912

10 37589119890 minus 10

(17479119890 minus 09)85629119890 minus 24

(19031119890 minus 23)52910119890 minus 50

(23037119890 minus 49)15947119890 minus 52

(14618119890 minus 51)89632119890 minus 146

(35734119890 minus 145)30640119890 minus 113

(30622119890 minus 112)

30 378269(313869)

00005(00007)

10278119890 minus 11

(20979119890 minus 11)11937119890 minus 09

(65345119890 minus 09)81119119890 minus 45

(44882119890 minus 44)30047119890 minus 10

(21285119890 minus 09)

50 68495119890 + 02

(10535119890 + 02)28041(22863)

00004(00010)

01214(06006)

24666119890 minus 15

(24614119890 minus 14)00223(00835)

70 14952119890 + 03

(18073119890 + 02)12961119890 + 02

(14505119890 + 02)02601(02405)

21146(39189)

12547119890 minus 05

(00001)11879(20240)

1198913

10 00013(00018)

40624119890 minus 08

(11066119890 minus 07)31542119890 minus 15

(19686119890 minus 14)36072119890 minus 27

(16376119890 minus 26)60846119890 minus 60

(58552119890 minus 59)44092119890 minus 51

(31687119890 minus 50)

30 20647119890 + 01

(79831)37596(20255)

05809(03373)

00091(00197)

27957119890 minus 06

(50455119890 minus 06)00258(00385)

50 72983119890 + 01

(28263119890 + 01)26189119890 + 01

(13125119890 + 01)10295119890 + 01

(51265)04883(03021)

00775(00620)

05720(03509)

70 14821119890 + 02

(46282119890 + 01)69337119890 + 01

(33718119890 + 01)29336119890 + 01

(10417119890 + 01)27039(10853)

11214(04917)

22342(07423)

1198914

10 27409119890 minus 09

(15090119890 minus 08)13708119890 minus 22

(65937119890 minus 22)35012119890 minus 49

(28064119890 minus 48)99156119890 minus 52

(69209119890 minus 51)16140119890 minus 144

(14296119890 minus 143)30561119890 minus 115

(21383119890 minus 114)

30 19825119890 + 02

(17202119890 + 02)00025(00029)

73319119890 minus 11

(23165119890 minus 10)66962119890 minus 09

(36744119890 minus 08)10828119890 minus 42

(89052119890 minus 42)56960119890 minus 08

(39998119890 minus 07)

50 32922119890 + 03

(56799119890 + 02)13397119890 + 01

(94756)00015(00019)

03064(12614)

22477119890 minus 17

(14523119890 minus 16)02379(11928)

70 74756119890 + 03

(75131119890 + 02)68348119890 + 02

(61444119890 + 02)16799(29749)

10304119890 + 01

(14839119890 + 01)19407119890 minus 06

(96092119890 minus 06)55072

(10286119890 + 01)

1198915

10 56418(1092)

43037(12401)

31701(12637)

07262(09490)

06503(14786)

06445(14706)

30 12322119890 + 03

(94694119890 + 02)42313119890 + 01

(26596119890 + 01)31052119890 + 01

(17699119890 + 01)23370119890 + 01

(19238)14702119890 + 01

(31506)19234119890 + 01

(29573)

50 65575119890 + 03

(13919119890 + 03)29055119890 + 02

(16229119890 + 02)73894119890 + 01

(36633119890 + 01)46563119890 + 01

(25861)374782119890 + 01

(27904)47741119890 + 01

(106763)

70 11834119890 + 04

(19184119890 + 03)52635119890 + 03

(40246119890 + 03)20414119890 + 02

(74873119890 + 01)75240119890 + 01

(12063119890 + 01)65838119890 + 01

(15847119890 + 01)84068119890 + 01

(30477119890 + 01)

1198916

10 51090(30667)

36806(17685)

38703(17882)

09750(09693)

22685(13272)

06765(08231)

30 15696119890 + 02

(39880119890 + 01)32291119890 + 01

(10059119890 + 01)29949119890 + 01

(71988)17282119890 + 01

(54892)18148119890 + 01

(62546)13312119890 + 01

(44942)

50 38006119890 + 02

(35258119890 + 01)75141119890 + 01

(2587119890 + 01)61896119890 + 01

(15107119890 + 01)36575119890 + 01

(94137)36236119890 + 01

(91136)30739119890 + 01

(85709)

70 58512119890 + 02

(36592119890 + 01)16259119890 + 02

(62498119890 + 01)88994119890 + 01

(17552119890 + 01)54198119890 + 01

(11482119890 + 01)52752119890 + 01

(12361119890 + 01)49043119890 + 01

(11607119890 + 01)

1198917

10 98646119890 minus 05

(00009)00041(00096)

00017(00055)

0(0)

0(0)

0(0)

30 01366(01090)

00008(00045)

00013(00071)

10210119890 minus 10

(10021119890 minus 09)35527119890 minus 17

(14708119890 minus 16)17067119890 minus 11

(13918119890 minus 10)

50 05856(00891)

00027(00021)

99034119890 minus 05

(00009)00001(00007)

24547119890 minus 15

(88945119890 minus 15)36456119890 minus 05

(00001)

70 06941(00646)

00329(00190)

00002(00010)

00011(00017)

44137119890 minus 11

(27956119890 minus 10)00007(00012)

Mathematical Problems in Engineering 7

Table 2 Continued

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198918

10 58516119890 minus 18

(25897119890 minus 17)22299119890 minus 40

(16366119890 minus 39)20832119890 minus 84

(16585119890 minus 83)25229119890 minus 87

(11045119890 minus 86)58546119890 minus 237

(0)22665119890 minus 154

(22664119890 minus 153)

30 19895119890 + 02

(62708119890 + 02)10418119890 minus 06

(37845119890 minus 06)73163119890 minus 18

(27837119890 minus 17)10483119890 minus 39

(72123119890 minus 39)30968119890 minus 99

(30467119890 minus 98)10029119890 minus 41

(10029119890 minus 40)

50 19927119890 + 07

(71603119890 + 07)18348119890 + 01

(48704119890 + 01)00017(00067)

65668119890 minus 25

(31648119890 minus 24)15985119890 minus 47

(14843119890 minus 46)37443119890 minus 23

(37143119890 minus 22)

70 15637119890 + 13

(83656119890 + 13)26886119890 + 08

(16837119890 + 09)14973119890 + 04

(10807119890 + 05)49509119890 minus 19

(26306119890 minus 18)32094119890 minus 29

(25130119890 minus 28)19384119890 minus 19

(10414119890 minus 18)

1198919

10 00472(01618)

01115(02369)

01773(02354)

50448119890 minus 15

(13412119890 minus 15)01269(02045)

00132(00770)

30 15006(02835)

05798(00888)

05469(00765)

03709(00736)

04187(00747)

03822(00767)

50 18625(01067)

06039(01209)

05348(00655)

03691(00544)

03848(00541)

03631(00478)

70 18765(00728)

08228(02349)

04790(00512)

03439(00463)

03369(00469)

03393(00413)

11989110

10 94158119890 minus 13

(49298119890 minus 12)14531119890 minus 26

(39844119890 minus 26)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)25554119890 minus 33

(10557119890 minus 32)

30 00527(00668)

34402119890 minus 07

(54202119890 minus 07)99564119890 minus 15

(20677119890 minus 14)67912119890 minus 12

(46051119890 minus 11)33310119890 minus 31

(14574119890 minus 30)83251119890 minus 13

(42490119890 minus 12)

50 13060(04239)

00010(00008)

18317119890 minus 07

(34350119890 minus 07)00003(00015)

00018(00127)

00011(00091)

70 26961(04601)

00350(00393)

00011(00091)

00059(00115)

00009(00090)

00023(00094)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0Set the coefficients as 119886 = 20 119887 = 02 119888 = 2 sdot 120587

(10) Penalised function one has

11989110 (119909) = 01sin2 (31205871199091) +

119863minus1

sum

119894=1

(119909119894minus 1)2[1 + sin2 (3120587119909

1)]

+ (119909119863minus 1)2[1 + sin2 (2120587119909

119863)]

+

119863

sum

119894=1

119906 (119909119894 5 100 4)

(23)

where

119906 (119909119894 119886 119896 119898) =

119896(119909119894minus 119886)119898 119909

119894gt 119886

0 minus119886 le 119909119894le 119886

119896(minus119909119894minus 119886)119898 119909119894lt minus119886

119910119894= 1 +

1

4

(119909119894+ 1)

(24)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [1 1]119863 with 119891(119909) = 0

42 The Coefficients Setting The parameters for simulationare listed in Table 1

In this table 119888(sdot)

expresses the accelerations coefficients120596 denotes inertia weight the dimension is 119863 and the rangeof the search space and the velocity space are 119909

(sdot)and V(sdot) If

the current position is out of the search space the positionof the particle is taken to be the value of the boundaryand the velocity is taken to be zero If the velocity of theparticle is outside of the boundary its value is set to bethe boundary value The maximum number of iterations isset to 1000 For each function 100 trials were carried outand the average optimal value and the standard deviationare presented To verify the performance of the algorithm atdifferent dimensions variable dimension 119863 increases from10 to 100 and the optimal mean and variance of benchmarkfunctions are calculated The results are presented in Table 2

5 The Results in Comparison withthe Previous Developments

The simulation results are given in Table 2 The comparisonresults elucidate that the searching accuracy and stabilityranging from low to high are listed as BPSO LWPSO EPSOTVAC ECPSO and CPSO for unimodal function It isobvious that the performances of ECPSO and CPSO aresuperior due to their advantage of obtaining the optimalspeed direction and the searching efficiency while in themultimodal function the CPSO algorithm is easy to trap intolocal minimum and TVAC shows better performance thanCPSO Combining the advantages of CPSO and TVAC theEPSO algorithm is applied to enhance the global search in

8 Mathematical Problems in Engineering

0 200 400 600 800 1000

0

05

1

15

2

25

3

35

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

0 200 400 600 800 1000

0

2

4

6

8

10

12

14

16

18

Number of generations

Mea

n be

st va

lue

0 200 400 600 800 1000

0

1

2

3

4

5

Number of generations

Number of generations Number of generations

Number of generations

0 200 400 600 800 1000

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000

0

100

200

300

400

500

0 200 400 600 800 1000

0

10

20

30

40

50

60

minus05

minus10

minus10minus100

minus1

minus2

f1

f5

f4f3

f2

f6

Figure 1 Continued

Mathematical Problems in Engineering 9

0 200 400 600 800 1000

0

005

01

015

02

025

03

035

Number of generations

Number of generations Number of generations

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

0 200 400 600 800 1000

0

05

1

15

2

25

3

0 200 400 600 800 1000

0

05

1

15

2

25

0 200 400 600 800 1000

0

002

004

006

008

01

012

014

016

minus05

minus05

minus005

minus002

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

f8f7

f10f9

Figure 1 Variation of the average optimum value with time

the early part of the optimization and encourage the particlesto converge toward the global optima at the end of the searchComparing to TVAC andCPSO the proposed new algorithmECPSO is appropriate for multimodal function search

With the increase of benchmark functions dimensionthe searching accuracy and stability for each algorithmare decreased The performances of CPSO and ECPSO aresuperior to the other algorithms With regard to multimodalfunction the performances of ECPSOandTVACare superiorto CPSO The average fitness is varied as in Table 2 FromFigure 1 it shows that the order of searching speed fromhigh to low is BPSO (green solid curve) LWPSO (red circle)EPSO (blue triangle) TVAC (black dot curve) CPSO (bluedash-dot) and ECPSO (purples dash) It is obvious that theperformances of EPSO and CPSO are more effective than theother algorithms

6 Calibrate the UnderwaterTransponder Coordinates

After two decades of dedicated research and developmentunmanned underwater vehicles (UUV) have been acceptedby an increasing number of users in bothmilitary and civilianinstitutions The design and implementation of navigationsystems stand out as one of the most critical steps towardsthe successful operation of autonomous vehicles The qualityof the overall estimates of the navigation system dramaticallyinfluences the capability of the vehicles to perform precision-demanding tasks [32] From a navigation point of view asingle range transponder may be regarded as an underwaterlighthouse providingUUVwith the ranges relative to its fixedgeographical location Single transponder navigation is not anew concept As mentioned the first at-sea demonstration of

10 Mathematical Problems in Engineering

Table 3 The calibration of the underwater transponder coordinates

BPSO LWPSO EPSO TVAC CPSO ECPSO1198901 35371119890 minus 15 minus23573119890 minus 15 26542119890 minus 15 minus19813119890 minus 15 minus11700119890 minus 15 14526119890 minus 15

1198902 15510119890 minus 15 23280119890 minus 15 minus48804119890 minus 16 minus14289119890 minus 15 13702119890 minus 15 22549119890 minus 15

1198903 0 0 0 0 0 0119904119902 41948119890 minus 15 27057119890 minus 15 26986119890 minus 15 24428119890 minus 15 18018119890 minus 15 26823119890 minus 15

Layingpoint

z

yG

x

Surfaceship

Transponder

Seabed

Figure 2 The calibration geometry for calibration of the transpon-der position

single transponder UTP aided inertial navigation was carriedout in 2003 as described in [33]

For ranging techniques like UTP to work the geo-graphical location of the transponders must be known Thepreferred method is to measure the position directly usingUSBL on a surface ship When the transponder deploymentis completed a surface ship with USBL and GPS sails aroundthe transponder in a circular motion collecting surface shipposition and distance information The calibration geometryis illustrated in Figure 2

In order to set the design framework let 119866 denote theglobal coordinate frame and let 119861 denote a coordinateframe attached to the vehicle usually denominated as body-fixed coordinate frameThe frame 119874 is the transceiver coor-dinates The position of transponder in the global coordinateframe is given by

119866119883119879=119866119883119863minus

119866

119861119877119861119883119863+119866

119861119877(

119861

119874119877119874119883119879+119861119883119874) (25)

where 119866119883119863is the position of the GPS in global coordinates

119861119883119863is the position of the GPS in vehicle coordinates 119874119883

119879

is the position of the transponder in transceiver coordinates119861119883119874is the position of the transceiver in vehicle coordinates

119866

119861119877is the rotation matrix from 119861 to 119866 and 119861

119874119877is the

rotation matrix from 119874 to 119861After the data collection we can get the distance between

the transponder and transceiver and the corresponding GPSposition in the global coordinates The attitude of the vehiclecan be measured by the heading sensors and the pitchrollsensor so 119866

119861119877is known We also know 119861119883

119863 119861119883119874 and 119861

119874119877

According to (25) we have

119874119883119879=

119861

119874119877minus1(

119866

119861119877minus1(119866119883119879minus119866119883119863) +119861119883119863minus119861119883119874) (26)

We denote 119874119883119879119894= [119874119909119879119894

119874119910119879119894

119874119911119879119894]

119879

(119894 = 1 2 119873)

as the position of the transponder in the transceiver coor-dinates in the 119894th measurement point The cost functionbecomes

119865 =

119873

sum

119894=1

(radic119874

1199092

119879119894+

119874

1199102

119879119894+

119874

1199112

119879119894minus 119871119894)

2

119873

(27)

Particles optimization algorithms have been introducedDefining each particle as a coordinate of the transponderTheparameters of particle swarmoptimization for simulationare the same as in Section 4 And a surface ship moves ina circular motion with a radius of 100 the real coordinatesof transponder are 119866119883

119903119879= [0 0 100]

119879 To simplify theproblem 119866

119861119877is unit matrix and 119861119883

119863minus119861119883119874= [0 0 0]

119879and ignore the sensor measurement error The simulationdata is shown in Table 3 Obviously the particle swarmalgorithm can search for the transponder coordinates andobtain accurate results At the same time CPSO shows thebest performance

Remark 2 One has

1198901 =119866119909119879minus119866119909119903119879 1198902 =

119866119910119879minus119866119910119903119879

1198903 =119866119911119879minus119866119911119903119879 119904119902 = radic1198901

2+ 11989022+ 11989032

(28)

7 Conclusion

In this paper a novel strategy to improve the performance ofparticle swarm optimization is proposed to apply in calibra-tion of the underwater transponder coordinates To improvethe population-based search optimization algorithm eachparticle is evolved along two different directions to generatetwo homologous particles The cost functions of two homol-ogous particles are calculated to keep the optimal one andto eliminate the poor one Then the next generation particleis updated It is regarded as CPSO Ten classify benchmarkfunctions are introduced to reflect the effectiveness of theproposed algorithmThe simulation results demonstrate thatthe unimodal function CPSO algorithm is superior toBPSO LWPSO EPSO and TVAC on the searching accuracystability and convergence speed However considering themultimodal function the performance of TVAC is superiorto CPSO

Mathematical Problems in Engineering 11

Secondly to further improve the performance of CPSOthe ECPSO is proposed by combining the CPSO and theTVAC In the initial period of the evolution the individualexperience is a significant aspect with larger accelerationcoefficient and in the final period the swarm experience issuperior with a greater acceleration coefficient Simultane-ously the evolution for each particle at any time is towardstwo different inertia directions to generate two homologousparticles and to obtain its next generation particles Thesimulations show the effectiveness of multimodal function byusing ECPSOWith the incensement of benchmark functionsdimensions the accuracy and stability of each algorithm willdecrease but CPSO and EPSO display the best performance

At last the strategy to calibrate the underwater transpon-der coordinates using particle swarm algorithm is intro-duced As the cost function for transponder coordinates is aunimodal function CPSO shows better performance than theother algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the Natural Science Foun-dation of China (51179038 1109043 51309067E091002) andthe Program of New Century Excellent Talents in University(NCET-10-0053)

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the 1995 IEEE International Conference on NeuralNetworks vol 4 no 2 pp 1942ndash1948 December 1995

[2] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[3] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[4] M S Arumugam M V C Rao and A W C Tan ldquoA noveland effective particle swarm optimization like algorithm withextrapolation techniquerdquo Applied Soft Computing Journal vol9 no 1 pp 308ndash320 2009

[5] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 1999

[6] G Chen X Huang J Jia et al ldquoNatural exponential inertiaweight strategy in particle swarm optimizationrdquo in Proceedingsof the 6thWorld Congress on Intelligent Control and Automation(WCICA rsquo06) vol 1 pp 3672ndash3675 IEEE June 2006

[7] Q Ni and J Deng ldquoA new logistic dynamic particle swarm opti-mization algorithm based on random topologyrdquo The ScientificWorld Journal vol 2013 Article ID 409167 8 pages 2013

[8] M M Noel ldquoA new gradient based particle swarm optimiza-tion algorithm for accurate computation of global minimumrdquoApplied Soft Computing Journal vol 12 no 1 pp 353ndash359 2012

[9] M G Epitropakis V P Plagianakos and M N VrahatisldquoEvolving cognitive and social experience in particle swarmoptimization through differential evolutionrdquo in Proceedings ofthe IEEE Congress on Evolutionary Computation (CEC rsquo10) pp1ndash8 IEEE July 2010

[10] A A Mousa M A El-Shorbagy and W F Abd-El-WahedldquoLocal search based hybrid particle swarm optimization algo-rithm for multiobjective optimizationrdquo Swarm and Evolution-ary Computation vol 3 pp 1ndash14 2012

[11] M H Moradi and M Abedini ldquoA combination of genetic algo-rithm and particle swarm optimization for optimal DG locationand sizing in distribution systemsrdquo International Journal ofElectrical Power and Energy Systems vol 34 no 1 pp 66ndash742012

[12] W D Chang J P Cheng M C Hsu et al ldquoParameteridentification of nonlinear systems using a particle swarmoptimization approachrdquo in Proceedings of the 3rd InternationalConference on Networking and Computing (ICNC rsquo12) pp 113ndash117 IEEE 2012

[13] J J Soon and K S Low ldquoPhotovoltaic model identificationusing particle swarm optimization with inverse barrier con-straintrdquo IEEE Transactions on Power Electronics vol 27 no 9pp 3975ndash3983 2012

[14] B Jiang N Wang and L Wang ldquoParameter identification forsolid oxide fuel cells using cooperative barebone particle swarmoptimization with hybrid learningrdquo International Journal ofHydrogen Energy vol 39 no 1 pp 532ndash542 2013

[15] A Alfi ldquoParticle swarm optimization algorithm with DynamicInertia Weight for online parameter identification applied toLorenz chaotic systemrdquo International Journal of InnovativeComputing Information and Control vol 8 no 2 pp 1191ndash12042012

[16] X Hu S Shi and X Gu ldquoAn improved particle swarm opti-mization algorithm for wireless sensor networks localizationrdquoin Proceedings of the 8th International Conference on WirelessCommunications Networking and Mobile Computing (WiCOMrsquo12) pp 1ndash4 Shanghai China September 2012

[17] F Sun J Yu and D Xu ldquoVisual measurement and control forunderwater robots a surveyrdquo in Proceedings of the 25th ChineseControl andDecision Conference (CCDC rsquo13) pp 333ndash338 IEEE2013

[18] V Djapic D Nad G Ferri et al ldquoNovel method for underwaternavigation aiding using a companion underwater robot asa guiding platformsrdquo in Proceedings of the 2013 MTSIEEEOCEANS-Bergen pp 1ndash10 IEEE June 2013

[19] A Caiti V Calabro D Meucci et al ldquoUnderwater Robots PastPresent and Futurerdquo 422ndash437

[20] P Krishnamurthy and F Khorrami ldquoA self-aligning underwaternavigation system based on fusion ofmultiple sensors includingDVL and IMUrdquo in Proceedings of the 9th Asian Control Confer-ence (ASCC rsquo13) pp 1ndash6 IEEE 2013

[21] S Wang and F Kang ldquoResearch and design of UUV navigationand control integrative simulation system based on compo-nentrdquo Intelligent Information Management vol 4 no 5 pp 181ndash187 2012

[22] T Kashima A Asada and T Ura ldquoThe positioning sys-tem integrated LBL and SSBL using seafloor acoustic mirrortransponderrdquo in Proceedings of the IEEE International Underwa-ter Technology Symposium (UT rsquo13) March 2013

12 Mathematical Problems in Engineering

[23] P Batista C Silvestre and P Oliveira ldquoGAS tightly coupledLBLUSBL position and velocity filter for underwater vehiclesrdquoin Proceedings of the European Control Conference (ECC rsquo13) pp2982ndash2987 Zurich Switzerland 2013

[24] M Morgado P Oliveira and C Silvestre ldquoTightly coupledultrashort baseline and inertial navigation system for under-water vehicles an experimental validationrdquo Journal of FieldRobotics vol 30 no 1 pp 142ndash170 2013

[25] Y U Min and H Junyin ldquoThe calibration of the USBL trans-ducer array for Long-range precision underwater positioningrdquoin Proceedings of the IEEE 10th International Conference onSignal Processing (ICSP rsquo10) pp 2357ndash2360 IEEE October 2010

[26] D Ji Y Li and J Liu ldquoSeafloor transponder calibration usingimproved perpendiculars intersectionrdquoAppliedOceanResearchvol 32 no 3 pp 261ndash266 2010

[27] B Jiao Z Lian and Q Chen ldquoA dynamic global and local com-bined particle swarm optimization algorithmrdquo Chaos Solitonsand Fractals vol 42 no 5 pp 2688ndash2695 2009

[28] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 IEEE May 1998

[29] M Schwaab E C Biscaia Jr J L Monteiro and J CPinto ldquoNonlinear parameter estimation through particle swarmoptimizationrdquo Chemical Engineering Science vol 63 no 6 pp1542ndash1552 2008

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII pp591ndash600 Springer Berlin Germany 1998

[31] A Chander A Chatterjee and P Siarry ldquoA new social andmomentum component adaptive PSO algorithm for imagesegmentationrdquo Expert Systems with Applications vol 38 no 5pp 4998ndash5004 2011

[32] M Morgado P Batista P Oliveira and C Silvestre ldquoPositionUSBLDVL sensor-based navigation filter in the presence ofunknown ocean currentsrdquoAutomatica vol 47 no 12 pp 2604ndash2614 2011

[33] B Jalving K Gade O K Hagen and K Vestgard ldquoA toolboxof aiding techniques for the HUGIN AUV integrated inertialnavigation systemrdquo in Proceedings of the OCEANS 2003 vol 2pp 1146ndash1153 IEEE September 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table 2 Comparison of BPSO WPSO EPSO TVAC and CPSO

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198911

10 47979119890 minus 11

(11343119890 minus 10)32219119890 minus 24

(10706119890 minus 23)87747119890 minus 51

(3408119890 minus 50)14826119890 minus 54

(14770119890 minus 53)41023119890 minus 146

(36388119890 minus 145)41221119890 minus 118

(30916119890 minus 117)

30 53768(43459)

40137119890 minus 05

(5926119890 minus 05)23219119890 minus 12

(11495119890 minus 11)11062119890 minus 10

(73302119890 minus 10)14872119890 minus 46

(64922119890 minus 46)16829119890 minus 10

(10243119890 minus 09)

50 28412119890 + 01

(45047)02149(01634)

22215119890 minus 05

(35696119890 minus 05)00019(00079)

93999119890 minus 20

(40558119890 minus 19)00022(00083)

70 44686119890 + 01

(43454)11728119890 + 01

(12113119890 + 01)00192(00238)

00609(01033)

20133119890 minus 07

(19424119890 minus 06)00302(00458)

1198912

10 37589119890 minus 10

(17479119890 minus 09)85629119890 minus 24

(19031119890 minus 23)52910119890 minus 50

(23037119890 minus 49)15947119890 minus 52

(14618119890 minus 51)89632119890 minus 146

(35734119890 minus 145)30640119890 minus 113

(30622119890 minus 112)

30 378269(313869)

00005(00007)

10278119890 minus 11

(20979119890 minus 11)11937119890 minus 09

(65345119890 minus 09)81119119890 minus 45

(44882119890 minus 44)30047119890 minus 10

(21285119890 minus 09)

50 68495119890 + 02

(10535119890 + 02)28041(22863)

00004(00010)

01214(06006)

24666119890 minus 15

(24614119890 minus 14)00223(00835)

70 14952119890 + 03

(18073119890 + 02)12961119890 + 02

(14505119890 + 02)02601(02405)

21146(39189)

12547119890 minus 05

(00001)11879(20240)

1198913

10 00013(00018)

40624119890 minus 08

(11066119890 minus 07)31542119890 minus 15

(19686119890 minus 14)36072119890 minus 27

(16376119890 minus 26)60846119890 minus 60

(58552119890 minus 59)44092119890 minus 51

(31687119890 minus 50)

30 20647119890 + 01

(79831)37596(20255)

05809(03373)

00091(00197)

27957119890 minus 06

(50455119890 minus 06)00258(00385)

50 72983119890 + 01

(28263119890 + 01)26189119890 + 01

(13125119890 + 01)10295119890 + 01

(51265)04883(03021)

00775(00620)

05720(03509)

70 14821119890 + 02

(46282119890 + 01)69337119890 + 01

(33718119890 + 01)29336119890 + 01

(10417119890 + 01)27039(10853)

11214(04917)

22342(07423)

1198914

10 27409119890 minus 09

(15090119890 minus 08)13708119890 minus 22

(65937119890 minus 22)35012119890 minus 49

(28064119890 minus 48)99156119890 minus 52

(69209119890 minus 51)16140119890 minus 144

(14296119890 minus 143)30561119890 minus 115

(21383119890 minus 114)

30 19825119890 + 02

(17202119890 + 02)00025(00029)

73319119890 minus 11

(23165119890 minus 10)66962119890 minus 09

(36744119890 minus 08)10828119890 minus 42

(89052119890 minus 42)56960119890 minus 08

(39998119890 minus 07)

50 32922119890 + 03

(56799119890 + 02)13397119890 + 01

(94756)00015(00019)

03064(12614)

22477119890 minus 17

(14523119890 minus 16)02379(11928)

70 74756119890 + 03

(75131119890 + 02)68348119890 + 02

(61444119890 + 02)16799(29749)

10304119890 + 01

(14839119890 + 01)19407119890 minus 06

(96092119890 minus 06)55072

(10286119890 + 01)

1198915

10 56418(1092)

43037(12401)

31701(12637)

07262(09490)

06503(14786)

06445(14706)

30 12322119890 + 03

(94694119890 + 02)42313119890 + 01

(26596119890 + 01)31052119890 + 01

(17699119890 + 01)23370119890 + 01

(19238)14702119890 + 01

(31506)19234119890 + 01

(29573)

50 65575119890 + 03

(13919119890 + 03)29055119890 + 02

(16229119890 + 02)73894119890 + 01

(36633119890 + 01)46563119890 + 01

(25861)374782119890 + 01

(27904)47741119890 + 01

(106763)

70 11834119890 + 04

(19184119890 + 03)52635119890 + 03

(40246119890 + 03)20414119890 + 02

(74873119890 + 01)75240119890 + 01

(12063119890 + 01)65838119890 + 01

(15847119890 + 01)84068119890 + 01

(30477119890 + 01)

1198916

10 51090(30667)

36806(17685)

38703(17882)

09750(09693)

22685(13272)

06765(08231)

30 15696119890 + 02

(39880119890 + 01)32291119890 + 01

(10059119890 + 01)29949119890 + 01

(71988)17282119890 + 01

(54892)18148119890 + 01

(62546)13312119890 + 01

(44942)

50 38006119890 + 02

(35258119890 + 01)75141119890 + 01

(2587119890 + 01)61896119890 + 01

(15107119890 + 01)36575119890 + 01

(94137)36236119890 + 01

(91136)30739119890 + 01

(85709)

70 58512119890 + 02

(36592119890 + 01)16259119890 + 02

(62498119890 + 01)88994119890 + 01

(17552119890 + 01)54198119890 + 01

(11482119890 + 01)52752119890 + 01

(12361119890 + 01)49043119890 + 01

(11607119890 + 01)

1198917

10 98646119890 minus 05

(00009)00041(00096)

00017(00055)

0(0)

0(0)

0(0)

30 01366(01090)

00008(00045)

00013(00071)

10210119890 minus 10

(10021119890 minus 09)35527119890 minus 17

(14708119890 minus 16)17067119890 minus 11

(13918119890 minus 10)

50 05856(00891)

00027(00021)

99034119890 minus 05

(00009)00001(00007)

24547119890 minus 15

(88945119890 minus 15)36456119890 minus 05

(00001)

70 06941(00646)

00329(00190)

00002(00010)

00011(00017)

44137119890 minus 11

(27956119890 minus 10)00007(00012)

Mathematical Problems in Engineering 7

Table 2 Continued

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198918

10 58516119890 minus 18

(25897119890 minus 17)22299119890 minus 40

(16366119890 minus 39)20832119890 minus 84

(16585119890 minus 83)25229119890 minus 87

(11045119890 minus 86)58546119890 minus 237

(0)22665119890 minus 154

(22664119890 minus 153)

30 19895119890 + 02

(62708119890 + 02)10418119890 minus 06

(37845119890 minus 06)73163119890 minus 18

(27837119890 minus 17)10483119890 minus 39

(72123119890 minus 39)30968119890 minus 99

(30467119890 minus 98)10029119890 minus 41

(10029119890 minus 40)

50 19927119890 + 07

(71603119890 + 07)18348119890 + 01

(48704119890 + 01)00017(00067)

65668119890 minus 25

(31648119890 minus 24)15985119890 minus 47

(14843119890 minus 46)37443119890 minus 23

(37143119890 minus 22)

70 15637119890 + 13

(83656119890 + 13)26886119890 + 08

(16837119890 + 09)14973119890 + 04

(10807119890 + 05)49509119890 minus 19

(26306119890 minus 18)32094119890 minus 29

(25130119890 minus 28)19384119890 minus 19

(10414119890 minus 18)

1198919

10 00472(01618)

01115(02369)

01773(02354)

50448119890 minus 15

(13412119890 minus 15)01269(02045)

00132(00770)

30 15006(02835)

05798(00888)

05469(00765)

03709(00736)

04187(00747)

03822(00767)

50 18625(01067)

06039(01209)

05348(00655)

03691(00544)

03848(00541)

03631(00478)

70 18765(00728)

08228(02349)

04790(00512)

03439(00463)

03369(00469)

03393(00413)

11989110

10 94158119890 minus 13

(49298119890 minus 12)14531119890 minus 26

(39844119890 minus 26)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)25554119890 minus 33

(10557119890 minus 32)

30 00527(00668)

34402119890 minus 07

(54202119890 minus 07)99564119890 minus 15

(20677119890 minus 14)67912119890 minus 12

(46051119890 minus 11)33310119890 minus 31

(14574119890 minus 30)83251119890 minus 13

(42490119890 minus 12)

50 13060(04239)

00010(00008)

18317119890 minus 07

(34350119890 minus 07)00003(00015)

00018(00127)

00011(00091)

70 26961(04601)

00350(00393)

00011(00091)

00059(00115)

00009(00090)

00023(00094)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0Set the coefficients as 119886 = 20 119887 = 02 119888 = 2 sdot 120587

(10) Penalised function one has

11989110 (119909) = 01sin2 (31205871199091) +

119863minus1

sum

119894=1

(119909119894minus 1)2[1 + sin2 (3120587119909

1)]

+ (119909119863minus 1)2[1 + sin2 (2120587119909

119863)]

+

119863

sum

119894=1

119906 (119909119894 5 100 4)

(23)

where

119906 (119909119894 119886 119896 119898) =

119896(119909119894minus 119886)119898 119909

119894gt 119886

0 minus119886 le 119909119894le 119886

119896(minus119909119894minus 119886)119898 119909119894lt minus119886

119910119894= 1 +

1

4

(119909119894+ 1)

(24)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [1 1]119863 with 119891(119909) = 0

42 The Coefficients Setting The parameters for simulationare listed in Table 1

In this table 119888(sdot)

expresses the accelerations coefficients120596 denotes inertia weight the dimension is 119863 and the rangeof the search space and the velocity space are 119909

(sdot)and V(sdot) If

the current position is out of the search space the positionof the particle is taken to be the value of the boundaryand the velocity is taken to be zero If the velocity of theparticle is outside of the boundary its value is set to bethe boundary value The maximum number of iterations isset to 1000 For each function 100 trials were carried outand the average optimal value and the standard deviationare presented To verify the performance of the algorithm atdifferent dimensions variable dimension 119863 increases from10 to 100 and the optimal mean and variance of benchmarkfunctions are calculated The results are presented in Table 2

5 The Results in Comparison withthe Previous Developments

The simulation results are given in Table 2 The comparisonresults elucidate that the searching accuracy and stabilityranging from low to high are listed as BPSO LWPSO EPSOTVAC ECPSO and CPSO for unimodal function It isobvious that the performances of ECPSO and CPSO aresuperior due to their advantage of obtaining the optimalspeed direction and the searching efficiency while in themultimodal function the CPSO algorithm is easy to trap intolocal minimum and TVAC shows better performance thanCPSO Combining the advantages of CPSO and TVAC theEPSO algorithm is applied to enhance the global search in

8 Mathematical Problems in Engineering

0 200 400 600 800 1000

0

05

1

15

2

25

3

35

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

0 200 400 600 800 1000

0

2

4

6

8

10

12

14

16

18

Number of generations

Mea

n be

st va

lue

0 200 400 600 800 1000

0

1

2

3

4

5

Number of generations

Number of generations Number of generations

Number of generations

0 200 400 600 800 1000

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000

0

100

200

300

400

500

0 200 400 600 800 1000

0

10

20

30

40

50

60

minus05

minus10

minus10minus100

minus1

minus2

f1

f5

f4f3

f2

f6

Figure 1 Continued

Mathematical Problems in Engineering 9

0 200 400 600 800 1000

0

005

01

015

02

025

03

035

Number of generations

Number of generations Number of generations

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

0 200 400 600 800 1000

0

05

1

15

2

25

3

0 200 400 600 800 1000

0

05

1

15

2

25

0 200 400 600 800 1000

0

002

004

006

008

01

012

014

016

minus05

minus05

minus005

minus002

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

f8f7

f10f9

Figure 1 Variation of the average optimum value with time

the early part of the optimization and encourage the particlesto converge toward the global optima at the end of the searchComparing to TVAC andCPSO the proposed new algorithmECPSO is appropriate for multimodal function search

With the increase of benchmark functions dimensionthe searching accuracy and stability for each algorithmare decreased The performances of CPSO and ECPSO aresuperior to the other algorithms With regard to multimodalfunction the performances of ECPSOandTVACare superiorto CPSO The average fitness is varied as in Table 2 FromFigure 1 it shows that the order of searching speed fromhigh to low is BPSO (green solid curve) LWPSO (red circle)EPSO (blue triangle) TVAC (black dot curve) CPSO (bluedash-dot) and ECPSO (purples dash) It is obvious that theperformances of EPSO and CPSO are more effective than theother algorithms

6 Calibrate the UnderwaterTransponder Coordinates

After two decades of dedicated research and developmentunmanned underwater vehicles (UUV) have been acceptedby an increasing number of users in bothmilitary and civilianinstitutions The design and implementation of navigationsystems stand out as one of the most critical steps towardsthe successful operation of autonomous vehicles The qualityof the overall estimates of the navigation system dramaticallyinfluences the capability of the vehicles to perform precision-demanding tasks [32] From a navigation point of view asingle range transponder may be regarded as an underwaterlighthouse providingUUVwith the ranges relative to its fixedgeographical location Single transponder navigation is not anew concept As mentioned the first at-sea demonstration of

10 Mathematical Problems in Engineering

Table 3 The calibration of the underwater transponder coordinates

BPSO LWPSO EPSO TVAC CPSO ECPSO1198901 35371119890 minus 15 minus23573119890 minus 15 26542119890 minus 15 minus19813119890 minus 15 minus11700119890 minus 15 14526119890 minus 15

1198902 15510119890 minus 15 23280119890 minus 15 minus48804119890 minus 16 minus14289119890 minus 15 13702119890 minus 15 22549119890 minus 15

1198903 0 0 0 0 0 0119904119902 41948119890 minus 15 27057119890 minus 15 26986119890 minus 15 24428119890 minus 15 18018119890 minus 15 26823119890 minus 15

Layingpoint

z

yG

x

Surfaceship

Transponder

Seabed

Figure 2 The calibration geometry for calibration of the transpon-der position

single transponder UTP aided inertial navigation was carriedout in 2003 as described in [33]

For ranging techniques like UTP to work the geo-graphical location of the transponders must be known Thepreferred method is to measure the position directly usingUSBL on a surface ship When the transponder deploymentis completed a surface ship with USBL and GPS sails aroundthe transponder in a circular motion collecting surface shipposition and distance information The calibration geometryis illustrated in Figure 2

In order to set the design framework let 119866 denote theglobal coordinate frame and let 119861 denote a coordinateframe attached to the vehicle usually denominated as body-fixed coordinate frameThe frame 119874 is the transceiver coor-dinates The position of transponder in the global coordinateframe is given by

119866119883119879=119866119883119863minus

119866

119861119877119861119883119863+119866

119861119877(

119861

119874119877119874119883119879+119861119883119874) (25)

where 119866119883119863is the position of the GPS in global coordinates

119861119883119863is the position of the GPS in vehicle coordinates 119874119883

119879

is the position of the transponder in transceiver coordinates119861119883119874is the position of the transceiver in vehicle coordinates

119866

119861119877is the rotation matrix from 119861 to 119866 and 119861

119874119877is the

rotation matrix from 119874 to 119861After the data collection we can get the distance between

the transponder and transceiver and the corresponding GPSposition in the global coordinates The attitude of the vehiclecan be measured by the heading sensors and the pitchrollsensor so 119866

119861119877is known We also know 119861119883

119863 119861119883119874 and 119861

119874119877

According to (25) we have

119874119883119879=

119861

119874119877minus1(

119866

119861119877minus1(119866119883119879minus119866119883119863) +119861119883119863minus119861119883119874) (26)

We denote 119874119883119879119894= [119874119909119879119894

119874119910119879119894

119874119911119879119894]

119879

(119894 = 1 2 119873)

as the position of the transponder in the transceiver coor-dinates in the 119894th measurement point The cost functionbecomes

119865 =

119873

sum

119894=1

(radic119874

1199092

119879119894+

119874

1199102

119879119894+

119874

1199112

119879119894minus 119871119894)

2

119873

(27)

Particles optimization algorithms have been introducedDefining each particle as a coordinate of the transponderTheparameters of particle swarmoptimization for simulationare the same as in Section 4 And a surface ship moves ina circular motion with a radius of 100 the real coordinatesof transponder are 119866119883

119903119879= [0 0 100]

119879 To simplify theproblem 119866

119861119877is unit matrix and 119861119883

119863minus119861119883119874= [0 0 0]

119879and ignore the sensor measurement error The simulationdata is shown in Table 3 Obviously the particle swarmalgorithm can search for the transponder coordinates andobtain accurate results At the same time CPSO shows thebest performance

Remark 2 One has

1198901 =119866119909119879minus119866119909119903119879 1198902 =

119866119910119879minus119866119910119903119879

1198903 =119866119911119879minus119866119911119903119879 119904119902 = radic1198901

2+ 11989022+ 11989032

(28)

7 Conclusion

In this paper a novel strategy to improve the performance ofparticle swarm optimization is proposed to apply in calibra-tion of the underwater transponder coordinates To improvethe population-based search optimization algorithm eachparticle is evolved along two different directions to generatetwo homologous particles The cost functions of two homol-ogous particles are calculated to keep the optimal one andto eliminate the poor one Then the next generation particleis updated It is regarded as CPSO Ten classify benchmarkfunctions are introduced to reflect the effectiveness of theproposed algorithmThe simulation results demonstrate thatthe unimodal function CPSO algorithm is superior toBPSO LWPSO EPSO and TVAC on the searching accuracystability and convergence speed However considering themultimodal function the performance of TVAC is superiorto CPSO

Mathematical Problems in Engineering 11

Secondly to further improve the performance of CPSOthe ECPSO is proposed by combining the CPSO and theTVAC In the initial period of the evolution the individualexperience is a significant aspect with larger accelerationcoefficient and in the final period the swarm experience issuperior with a greater acceleration coefficient Simultane-ously the evolution for each particle at any time is towardstwo different inertia directions to generate two homologousparticles and to obtain its next generation particles Thesimulations show the effectiveness of multimodal function byusing ECPSOWith the incensement of benchmark functionsdimensions the accuracy and stability of each algorithm willdecrease but CPSO and EPSO display the best performance

At last the strategy to calibrate the underwater transpon-der coordinates using particle swarm algorithm is intro-duced As the cost function for transponder coordinates is aunimodal function CPSO shows better performance than theother algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the Natural Science Foun-dation of China (51179038 1109043 51309067E091002) andthe Program of New Century Excellent Talents in University(NCET-10-0053)

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the 1995 IEEE International Conference on NeuralNetworks vol 4 no 2 pp 1942ndash1948 December 1995

[2] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[3] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[4] M S Arumugam M V C Rao and A W C Tan ldquoA noveland effective particle swarm optimization like algorithm withextrapolation techniquerdquo Applied Soft Computing Journal vol9 no 1 pp 308ndash320 2009

[5] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 1999

[6] G Chen X Huang J Jia et al ldquoNatural exponential inertiaweight strategy in particle swarm optimizationrdquo in Proceedingsof the 6thWorld Congress on Intelligent Control and Automation(WCICA rsquo06) vol 1 pp 3672ndash3675 IEEE June 2006

[7] Q Ni and J Deng ldquoA new logistic dynamic particle swarm opti-mization algorithm based on random topologyrdquo The ScientificWorld Journal vol 2013 Article ID 409167 8 pages 2013

[8] M M Noel ldquoA new gradient based particle swarm optimiza-tion algorithm for accurate computation of global minimumrdquoApplied Soft Computing Journal vol 12 no 1 pp 353ndash359 2012

[9] M G Epitropakis V P Plagianakos and M N VrahatisldquoEvolving cognitive and social experience in particle swarmoptimization through differential evolutionrdquo in Proceedings ofthe IEEE Congress on Evolutionary Computation (CEC rsquo10) pp1ndash8 IEEE July 2010

[10] A A Mousa M A El-Shorbagy and W F Abd-El-WahedldquoLocal search based hybrid particle swarm optimization algo-rithm for multiobjective optimizationrdquo Swarm and Evolution-ary Computation vol 3 pp 1ndash14 2012

[11] M H Moradi and M Abedini ldquoA combination of genetic algo-rithm and particle swarm optimization for optimal DG locationand sizing in distribution systemsrdquo International Journal ofElectrical Power and Energy Systems vol 34 no 1 pp 66ndash742012

[12] W D Chang J P Cheng M C Hsu et al ldquoParameteridentification of nonlinear systems using a particle swarmoptimization approachrdquo in Proceedings of the 3rd InternationalConference on Networking and Computing (ICNC rsquo12) pp 113ndash117 IEEE 2012

[13] J J Soon and K S Low ldquoPhotovoltaic model identificationusing particle swarm optimization with inverse barrier con-straintrdquo IEEE Transactions on Power Electronics vol 27 no 9pp 3975ndash3983 2012

[14] B Jiang N Wang and L Wang ldquoParameter identification forsolid oxide fuel cells using cooperative barebone particle swarmoptimization with hybrid learningrdquo International Journal ofHydrogen Energy vol 39 no 1 pp 532ndash542 2013

[15] A Alfi ldquoParticle swarm optimization algorithm with DynamicInertia Weight for online parameter identification applied toLorenz chaotic systemrdquo International Journal of InnovativeComputing Information and Control vol 8 no 2 pp 1191ndash12042012

[16] X Hu S Shi and X Gu ldquoAn improved particle swarm opti-mization algorithm for wireless sensor networks localizationrdquoin Proceedings of the 8th International Conference on WirelessCommunications Networking and Mobile Computing (WiCOMrsquo12) pp 1ndash4 Shanghai China September 2012

[17] F Sun J Yu and D Xu ldquoVisual measurement and control forunderwater robots a surveyrdquo in Proceedings of the 25th ChineseControl andDecision Conference (CCDC rsquo13) pp 333ndash338 IEEE2013

[18] V Djapic D Nad G Ferri et al ldquoNovel method for underwaternavigation aiding using a companion underwater robot asa guiding platformsrdquo in Proceedings of the 2013 MTSIEEEOCEANS-Bergen pp 1ndash10 IEEE June 2013

[19] A Caiti V Calabro D Meucci et al ldquoUnderwater Robots PastPresent and Futurerdquo 422ndash437

[20] P Krishnamurthy and F Khorrami ldquoA self-aligning underwaternavigation system based on fusion ofmultiple sensors includingDVL and IMUrdquo in Proceedings of the 9th Asian Control Confer-ence (ASCC rsquo13) pp 1ndash6 IEEE 2013

[21] S Wang and F Kang ldquoResearch and design of UUV navigationand control integrative simulation system based on compo-nentrdquo Intelligent Information Management vol 4 no 5 pp 181ndash187 2012

[22] T Kashima A Asada and T Ura ldquoThe positioning sys-tem integrated LBL and SSBL using seafloor acoustic mirrortransponderrdquo in Proceedings of the IEEE International Underwa-ter Technology Symposium (UT rsquo13) March 2013

12 Mathematical Problems in Engineering

[23] P Batista C Silvestre and P Oliveira ldquoGAS tightly coupledLBLUSBL position and velocity filter for underwater vehiclesrdquoin Proceedings of the European Control Conference (ECC rsquo13) pp2982ndash2987 Zurich Switzerland 2013

[24] M Morgado P Oliveira and C Silvestre ldquoTightly coupledultrashort baseline and inertial navigation system for under-water vehicles an experimental validationrdquo Journal of FieldRobotics vol 30 no 1 pp 142ndash170 2013

[25] Y U Min and H Junyin ldquoThe calibration of the USBL trans-ducer array for Long-range precision underwater positioningrdquoin Proceedings of the IEEE 10th International Conference onSignal Processing (ICSP rsquo10) pp 2357ndash2360 IEEE October 2010

[26] D Ji Y Li and J Liu ldquoSeafloor transponder calibration usingimproved perpendiculars intersectionrdquoAppliedOceanResearchvol 32 no 3 pp 261ndash266 2010

[27] B Jiao Z Lian and Q Chen ldquoA dynamic global and local com-bined particle swarm optimization algorithmrdquo Chaos Solitonsand Fractals vol 42 no 5 pp 2688ndash2695 2009

[28] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 IEEE May 1998

[29] M Schwaab E C Biscaia Jr J L Monteiro and J CPinto ldquoNonlinear parameter estimation through particle swarmoptimizationrdquo Chemical Engineering Science vol 63 no 6 pp1542ndash1552 2008

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII pp591ndash600 Springer Berlin Germany 1998

[31] A Chander A Chatterjee and P Siarry ldquoA new social andmomentum component adaptive PSO algorithm for imagesegmentationrdquo Expert Systems with Applications vol 38 no 5pp 4998ndash5004 2011

[32] M Morgado P Batista P Oliveira and C Silvestre ldquoPositionUSBLDVL sensor-based navigation filter in the presence ofunknown ocean currentsrdquoAutomatica vol 47 no 12 pp 2604ndash2614 2011

[33] B Jalving K Gade O K Hagen and K Vestgard ldquoA toolboxof aiding techniques for the HUGIN AUV integrated inertialnavigation systemrdquo in Proceedings of the OCEANS 2003 vol 2pp 1146ndash1153 IEEE September 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 2 Continued

119865 119863

Average (standard deviation)BPSO LWPSO EPSO TVAC CPSO ECPSO

1198918

10 58516119890 minus 18

(25897119890 minus 17)22299119890 minus 40

(16366119890 minus 39)20832119890 minus 84

(16585119890 minus 83)25229119890 minus 87

(11045119890 minus 86)58546119890 minus 237

(0)22665119890 minus 154

(22664119890 minus 153)

30 19895119890 + 02

(62708119890 + 02)10418119890 minus 06

(37845119890 minus 06)73163119890 minus 18

(27837119890 minus 17)10483119890 minus 39

(72123119890 minus 39)30968119890 minus 99

(30467119890 minus 98)10029119890 minus 41

(10029119890 minus 40)

50 19927119890 + 07

(71603119890 + 07)18348119890 + 01

(48704119890 + 01)00017(00067)

65668119890 minus 25

(31648119890 minus 24)15985119890 minus 47

(14843119890 minus 46)37443119890 minus 23

(37143119890 minus 22)

70 15637119890 + 13

(83656119890 + 13)26886119890 + 08

(16837119890 + 09)14973119890 + 04

(10807119890 + 05)49509119890 minus 19

(26306119890 minus 18)32094119890 minus 29

(25130119890 minus 28)19384119890 minus 19

(10414119890 minus 18)

1198919

10 00472(01618)

01115(02369)

01773(02354)

50448119890 minus 15

(13412119890 minus 15)01269(02045)

00132(00770)

30 15006(02835)

05798(00888)

05469(00765)

03709(00736)

04187(00747)

03822(00767)

50 18625(01067)

06039(01209)

05348(00655)

03691(00544)

03848(00541)

03631(00478)

70 18765(00728)

08228(02349)

04790(00512)

03439(00463)

03369(00469)

03393(00413)

11989110

10 94158119890 minus 13

(49298119890 minus 12)14531119890 minus 26

(39844119890 minus 26)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)14997119890 minus 33

(34384119890 minus 48)25554119890 minus 33

(10557119890 minus 32)

30 00527(00668)

34402119890 minus 07

(54202119890 minus 07)99564119890 minus 15

(20677119890 minus 14)67912119890 minus 12

(46051119890 minus 11)33310119890 minus 31

(14574119890 minus 30)83251119890 minus 13

(42490119890 minus 12)

50 13060(04239)

00010(00008)

18317119890 minus 07

(34350119890 minus 07)00003(00015)

00018(00127)

00011(00091)

70 26961(04601)

00350(00393)

00011(00091)

00059(00115)

00009(00090)

00023(00094)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [0 0]119863 with 119891(119909) = 0Set the coefficients as 119886 = 20 119887 = 02 119888 = 2 sdot 120587

(10) Penalised function one has

11989110 (119909) = 01sin2 (31205871199091) +

119863minus1

sum

119894=1

(119909119894minus 1)2[1 + sin2 (3120587119909

1)]

+ (119909119863minus 1)2[1 + sin2 (2120587119909

119863)]

+

119863

sum

119894=1

119906 (119909119894 5 100 4)

(23)

where

119906 (119909119894 119886 119896 119898) =

119896(119909119894minus 119886)119898 119909

119894gt 119886

0 minus119886 le 119909119894le 119886

119896(minus119909119894minus 119886)119898 119909119894lt minus119886

119910119894= 1 +

1

4

(119909119894+ 1)

(24)

With the search space 119909119894| minus100 lt 119909

119894lt 100 the global

minimum locates at 119909 = [1 1]119863 with 119891(119909) = 0

42 The Coefficients Setting The parameters for simulationare listed in Table 1

In this table 119888(sdot)

expresses the accelerations coefficients120596 denotes inertia weight the dimension is 119863 and the rangeof the search space and the velocity space are 119909

(sdot)and V(sdot) If

the current position is out of the search space the positionof the particle is taken to be the value of the boundaryand the velocity is taken to be zero If the velocity of theparticle is outside of the boundary its value is set to bethe boundary value The maximum number of iterations isset to 1000 For each function 100 trials were carried outand the average optimal value and the standard deviationare presented To verify the performance of the algorithm atdifferent dimensions variable dimension 119863 increases from10 to 100 and the optimal mean and variance of benchmarkfunctions are calculated The results are presented in Table 2

5 The Results in Comparison withthe Previous Developments

The simulation results are given in Table 2 The comparisonresults elucidate that the searching accuracy and stabilityranging from low to high are listed as BPSO LWPSO EPSOTVAC ECPSO and CPSO for unimodal function It isobvious that the performances of ECPSO and CPSO aresuperior due to their advantage of obtaining the optimalspeed direction and the searching efficiency while in themultimodal function the CPSO algorithm is easy to trap intolocal minimum and TVAC shows better performance thanCPSO Combining the advantages of CPSO and TVAC theEPSO algorithm is applied to enhance the global search in

8 Mathematical Problems in Engineering

0 200 400 600 800 1000

0

05

1

15

2

25

3

35

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

0 200 400 600 800 1000

0

2

4

6

8

10

12

14

16

18

Number of generations

Mea

n be

st va

lue

0 200 400 600 800 1000

0

1

2

3

4

5

Number of generations

Number of generations Number of generations

Number of generations

0 200 400 600 800 1000

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000

0

100

200

300

400

500

0 200 400 600 800 1000

0

10

20

30

40

50

60

minus05

minus10

minus10minus100

minus1

minus2

f1

f5

f4f3

f2

f6

Figure 1 Continued

Mathematical Problems in Engineering 9

0 200 400 600 800 1000

0

005

01

015

02

025

03

035

Number of generations

Number of generations Number of generations

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

0 200 400 600 800 1000

0

05

1

15

2

25

3

0 200 400 600 800 1000

0

05

1

15

2

25

0 200 400 600 800 1000

0

002

004

006

008

01

012

014

016

minus05

minus05

minus005

minus002

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

f8f7

f10f9

Figure 1 Variation of the average optimum value with time

the early part of the optimization and encourage the particlesto converge toward the global optima at the end of the searchComparing to TVAC andCPSO the proposed new algorithmECPSO is appropriate for multimodal function search

With the increase of benchmark functions dimensionthe searching accuracy and stability for each algorithmare decreased The performances of CPSO and ECPSO aresuperior to the other algorithms With regard to multimodalfunction the performances of ECPSOandTVACare superiorto CPSO The average fitness is varied as in Table 2 FromFigure 1 it shows that the order of searching speed fromhigh to low is BPSO (green solid curve) LWPSO (red circle)EPSO (blue triangle) TVAC (black dot curve) CPSO (bluedash-dot) and ECPSO (purples dash) It is obvious that theperformances of EPSO and CPSO are more effective than theother algorithms

6 Calibrate the UnderwaterTransponder Coordinates

After two decades of dedicated research and developmentunmanned underwater vehicles (UUV) have been acceptedby an increasing number of users in bothmilitary and civilianinstitutions The design and implementation of navigationsystems stand out as one of the most critical steps towardsthe successful operation of autonomous vehicles The qualityof the overall estimates of the navigation system dramaticallyinfluences the capability of the vehicles to perform precision-demanding tasks [32] From a navigation point of view asingle range transponder may be regarded as an underwaterlighthouse providingUUVwith the ranges relative to its fixedgeographical location Single transponder navigation is not anew concept As mentioned the first at-sea demonstration of

10 Mathematical Problems in Engineering

Table 3 The calibration of the underwater transponder coordinates

BPSO LWPSO EPSO TVAC CPSO ECPSO1198901 35371119890 minus 15 minus23573119890 minus 15 26542119890 minus 15 minus19813119890 minus 15 minus11700119890 minus 15 14526119890 minus 15

1198902 15510119890 minus 15 23280119890 minus 15 minus48804119890 minus 16 minus14289119890 minus 15 13702119890 minus 15 22549119890 minus 15

1198903 0 0 0 0 0 0119904119902 41948119890 minus 15 27057119890 minus 15 26986119890 minus 15 24428119890 minus 15 18018119890 minus 15 26823119890 minus 15

Layingpoint

z

yG

x

Surfaceship

Transponder

Seabed

Figure 2 The calibration geometry for calibration of the transpon-der position

single transponder UTP aided inertial navigation was carriedout in 2003 as described in [33]

For ranging techniques like UTP to work the geo-graphical location of the transponders must be known Thepreferred method is to measure the position directly usingUSBL on a surface ship When the transponder deploymentis completed a surface ship with USBL and GPS sails aroundthe transponder in a circular motion collecting surface shipposition and distance information The calibration geometryis illustrated in Figure 2

In order to set the design framework let 119866 denote theglobal coordinate frame and let 119861 denote a coordinateframe attached to the vehicle usually denominated as body-fixed coordinate frameThe frame 119874 is the transceiver coor-dinates The position of transponder in the global coordinateframe is given by

119866119883119879=119866119883119863minus

119866

119861119877119861119883119863+119866

119861119877(

119861

119874119877119874119883119879+119861119883119874) (25)

where 119866119883119863is the position of the GPS in global coordinates

119861119883119863is the position of the GPS in vehicle coordinates 119874119883

119879

is the position of the transponder in transceiver coordinates119861119883119874is the position of the transceiver in vehicle coordinates

119866

119861119877is the rotation matrix from 119861 to 119866 and 119861

119874119877is the

rotation matrix from 119874 to 119861After the data collection we can get the distance between

the transponder and transceiver and the corresponding GPSposition in the global coordinates The attitude of the vehiclecan be measured by the heading sensors and the pitchrollsensor so 119866

119861119877is known We also know 119861119883

119863 119861119883119874 and 119861

119874119877

According to (25) we have

119874119883119879=

119861

119874119877minus1(

119866

119861119877minus1(119866119883119879minus119866119883119863) +119861119883119863minus119861119883119874) (26)

We denote 119874119883119879119894= [119874119909119879119894

119874119910119879119894

119874119911119879119894]

119879

(119894 = 1 2 119873)

as the position of the transponder in the transceiver coor-dinates in the 119894th measurement point The cost functionbecomes

119865 =

119873

sum

119894=1

(radic119874

1199092

119879119894+

119874

1199102

119879119894+

119874

1199112

119879119894minus 119871119894)

2

119873

(27)

Particles optimization algorithms have been introducedDefining each particle as a coordinate of the transponderTheparameters of particle swarmoptimization for simulationare the same as in Section 4 And a surface ship moves ina circular motion with a radius of 100 the real coordinatesof transponder are 119866119883

119903119879= [0 0 100]

119879 To simplify theproblem 119866

119861119877is unit matrix and 119861119883

119863minus119861119883119874= [0 0 0]

119879and ignore the sensor measurement error The simulationdata is shown in Table 3 Obviously the particle swarmalgorithm can search for the transponder coordinates andobtain accurate results At the same time CPSO shows thebest performance

Remark 2 One has

1198901 =119866119909119879minus119866119909119903119879 1198902 =

119866119910119879minus119866119910119903119879

1198903 =119866119911119879minus119866119911119903119879 119904119902 = radic1198901

2+ 11989022+ 11989032

(28)

7 Conclusion

In this paper a novel strategy to improve the performance ofparticle swarm optimization is proposed to apply in calibra-tion of the underwater transponder coordinates To improvethe population-based search optimization algorithm eachparticle is evolved along two different directions to generatetwo homologous particles The cost functions of two homol-ogous particles are calculated to keep the optimal one andto eliminate the poor one Then the next generation particleis updated It is regarded as CPSO Ten classify benchmarkfunctions are introduced to reflect the effectiveness of theproposed algorithmThe simulation results demonstrate thatthe unimodal function CPSO algorithm is superior toBPSO LWPSO EPSO and TVAC on the searching accuracystability and convergence speed However considering themultimodal function the performance of TVAC is superiorto CPSO

Mathematical Problems in Engineering 11

Secondly to further improve the performance of CPSOthe ECPSO is proposed by combining the CPSO and theTVAC In the initial period of the evolution the individualexperience is a significant aspect with larger accelerationcoefficient and in the final period the swarm experience issuperior with a greater acceleration coefficient Simultane-ously the evolution for each particle at any time is towardstwo different inertia directions to generate two homologousparticles and to obtain its next generation particles Thesimulations show the effectiveness of multimodal function byusing ECPSOWith the incensement of benchmark functionsdimensions the accuracy and stability of each algorithm willdecrease but CPSO and EPSO display the best performance

At last the strategy to calibrate the underwater transpon-der coordinates using particle swarm algorithm is intro-duced As the cost function for transponder coordinates is aunimodal function CPSO shows better performance than theother algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the Natural Science Foun-dation of China (51179038 1109043 51309067E091002) andthe Program of New Century Excellent Talents in University(NCET-10-0053)

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the 1995 IEEE International Conference on NeuralNetworks vol 4 no 2 pp 1942ndash1948 December 1995

[2] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[3] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[4] M S Arumugam M V C Rao and A W C Tan ldquoA noveland effective particle swarm optimization like algorithm withextrapolation techniquerdquo Applied Soft Computing Journal vol9 no 1 pp 308ndash320 2009

[5] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 1999

[6] G Chen X Huang J Jia et al ldquoNatural exponential inertiaweight strategy in particle swarm optimizationrdquo in Proceedingsof the 6thWorld Congress on Intelligent Control and Automation(WCICA rsquo06) vol 1 pp 3672ndash3675 IEEE June 2006

[7] Q Ni and J Deng ldquoA new logistic dynamic particle swarm opti-mization algorithm based on random topologyrdquo The ScientificWorld Journal vol 2013 Article ID 409167 8 pages 2013

[8] M M Noel ldquoA new gradient based particle swarm optimiza-tion algorithm for accurate computation of global minimumrdquoApplied Soft Computing Journal vol 12 no 1 pp 353ndash359 2012

[9] M G Epitropakis V P Plagianakos and M N VrahatisldquoEvolving cognitive and social experience in particle swarmoptimization through differential evolutionrdquo in Proceedings ofthe IEEE Congress on Evolutionary Computation (CEC rsquo10) pp1ndash8 IEEE July 2010

[10] A A Mousa M A El-Shorbagy and W F Abd-El-WahedldquoLocal search based hybrid particle swarm optimization algo-rithm for multiobjective optimizationrdquo Swarm and Evolution-ary Computation vol 3 pp 1ndash14 2012

[11] M H Moradi and M Abedini ldquoA combination of genetic algo-rithm and particle swarm optimization for optimal DG locationand sizing in distribution systemsrdquo International Journal ofElectrical Power and Energy Systems vol 34 no 1 pp 66ndash742012

[12] W D Chang J P Cheng M C Hsu et al ldquoParameteridentification of nonlinear systems using a particle swarmoptimization approachrdquo in Proceedings of the 3rd InternationalConference on Networking and Computing (ICNC rsquo12) pp 113ndash117 IEEE 2012

[13] J J Soon and K S Low ldquoPhotovoltaic model identificationusing particle swarm optimization with inverse barrier con-straintrdquo IEEE Transactions on Power Electronics vol 27 no 9pp 3975ndash3983 2012

[14] B Jiang N Wang and L Wang ldquoParameter identification forsolid oxide fuel cells using cooperative barebone particle swarmoptimization with hybrid learningrdquo International Journal ofHydrogen Energy vol 39 no 1 pp 532ndash542 2013

[15] A Alfi ldquoParticle swarm optimization algorithm with DynamicInertia Weight for online parameter identification applied toLorenz chaotic systemrdquo International Journal of InnovativeComputing Information and Control vol 8 no 2 pp 1191ndash12042012

[16] X Hu S Shi and X Gu ldquoAn improved particle swarm opti-mization algorithm for wireless sensor networks localizationrdquoin Proceedings of the 8th International Conference on WirelessCommunications Networking and Mobile Computing (WiCOMrsquo12) pp 1ndash4 Shanghai China September 2012

[17] F Sun J Yu and D Xu ldquoVisual measurement and control forunderwater robots a surveyrdquo in Proceedings of the 25th ChineseControl andDecision Conference (CCDC rsquo13) pp 333ndash338 IEEE2013

[18] V Djapic D Nad G Ferri et al ldquoNovel method for underwaternavigation aiding using a companion underwater robot asa guiding platformsrdquo in Proceedings of the 2013 MTSIEEEOCEANS-Bergen pp 1ndash10 IEEE June 2013

[19] A Caiti V Calabro D Meucci et al ldquoUnderwater Robots PastPresent and Futurerdquo 422ndash437

[20] P Krishnamurthy and F Khorrami ldquoA self-aligning underwaternavigation system based on fusion ofmultiple sensors includingDVL and IMUrdquo in Proceedings of the 9th Asian Control Confer-ence (ASCC rsquo13) pp 1ndash6 IEEE 2013

[21] S Wang and F Kang ldquoResearch and design of UUV navigationand control integrative simulation system based on compo-nentrdquo Intelligent Information Management vol 4 no 5 pp 181ndash187 2012

[22] T Kashima A Asada and T Ura ldquoThe positioning sys-tem integrated LBL and SSBL using seafloor acoustic mirrortransponderrdquo in Proceedings of the IEEE International Underwa-ter Technology Symposium (UT rsquo13) March 2013

12 Mathematical Problems in Engineering

[23] P Batista C Silvestre and P Oliveira ldquoGAS tightly coupledLBLUSBL position and velocity filter for underwater vehiclesrdquoin Proceedings of the European Control Conference (ECC rsquo13) pp2982ndash2987 Zurich Switzerland 2013

[24] M Morgado P Oliveira and C Silvestre ldquoTightly coupledultrashort baseline and inertial navigation system for under-water vehicles an experimental validationrdquo Journal of FieldRobotics vol 30 no 1 pp 142ndash170 2013

[25] Y U Min and H Junyin ldquoThe calibration of the USBL trans-ducer array for Long-range precision underwater positioningrdquoin Proceedings of the IEEE 10th International Conference onSignal Processing (ICSP rsquo10) pp 2357ndash2360 IEEE October 2010

[26] D Ji Y Li and J Liu ldquoSeafloor transponder calibration usingimproved perpendiculars intersectionrdquoAppliedOceanResearchvol 32 no 3 pp 261ndash266 2010

[27] B Jiao Z Lian and Q Chen ldquoA dynamic global and local com-bined particle swarm optimization algorithmrdquo Chaos Solitonsand Fractals vol 42 no 5 pp 2688ndash2695 2009

[28] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 IEEE May 1998

[29] M Schwaab E C Biscaia Jr J L Monteiro and J CPinto ldquoNonlinear parameter estimation through particle swarmoptimizationrdquo Chemical Engineering Science vol 63 no 6 pp1542ndash1552 2008

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII pp591ndash600 Springer Berlin Germany 1998

[31] A Chander A Chatterjee and P Siarry ldquoA new social andmomentum component adaptive PSO algorithm for imagesegmentationrdquo Expert Systems with Applications vol 38 no 5pp 4998ndash5004 2011

[32] M Morgado P Batista P Oliveira and C Silvestre ldquoPositionUSBLDVL sensor-based navigation filter in the presence ofunknown ocean currentsrdquoAutomatica vol 47 no 12 pp 2604ndash2614 2011

[33] B Jalving K Gade O K Hagen and K Vestgard ldquoA toolboxof aiding techniques for the HUGIN AUV integrated inertialnavigation systemrdquo in Proceedings of the OCEANS 2003 vol 2pp 1146ndash1153 IEEE September 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 200 400 600 800 1000

0

05

1

15

2

25

3

35

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

0 200 400 600 800 1000

0

2

4

6

8

10

12

14

16

18

Number of generations

Mea

n be

st va

lue

0 200 400 600 800 1000

0

1

2

3

4

5

Number of generations

Number of generations Number of generations

Number of generations

0 200 400 600 800 1000

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000

0

100

200

300

400

500

0 200 400 600 800 1000

0

10

20

30

40

50

60

minus05

minus10

minus10minus100

minus1

minus2

f1

f5

f4f3

f2

f6

Figure 1 Continued

Mathematical Problems in Engineering 9

0 200 400 600 800 1000

0

005

01

015

02

025

03

035

Number of generations

Number of generations Number of generations

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

0 200 400 600 800 1000

0

05

1

15

2

25

3

0 200 400 600 800 1000

0

05

1

15

2

25

0 200 400 600 800 1000

0

002

004

006

008

01

012

014

016

minus05

minus05

minus005

minus002

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

f8f7

f10f9

Figure 1 Variation of the average optimum value with time

the early part of the optimization and encourage the particlesto converge toward the global optima at the end of the searchComparing to TVAC andCPSO the proposed new algorithmECPSO is appropriate for multimodal function search

With the increase of benchmark functions dimensionthe searching accuracy and stability for each algorithmare decreased The performances of CPSO and ECPSO aresuperior to the other algorithms With regard to multimodalfunction the performances of ECPSOandTVACare superiorto CPSO The average fitness is varied as in Table 2 FromFigure 1 it shows that the order of searching speed fromhigh to low is BPSO (green solid curve) LWPSO (red circle)EPSO (blue triangle) TVAC (black dot curve) CPSO (bluedash-dot) and ECPSO (purples dash) It is obvious that theperformances of EPSO and CPSO are more effective than theother algorithms

6 Calibrate the UnderwaterTransponder Coordinates

After two decades of dedicated research and developmentunmanned underwater vehicles (UUV) have been acceptedby an increasing number of users in bothmilitary and civilianinstitutions The design and implementation of navigationsystems stand out as one of the most critical steps towardsthe successful operation of autonomous vehicles The qualityof the overall estimates of the navigation system dramaticallyinfluences the capability of the vehicles to perform precision-demanding tasks [32] From a navigation point of view asingle range transponder may be regarded as an underwaterlighthouse providingUUVwith the ranges relative to its fixedgeographical location Single transponder navigation is not anew concept As mentioned the first at-sea demonstration of

10 Mathematical Problems in Engineering

Table 3 The calibration of the underwater transponder coordinates

BPSO LWPSO EPSO TVAC CPSO ECPSO1198901 35371119890 minus 15 minus23573119890 minus 15 26542119890 minus 15 minus19813119890 minus 15 minus11700119890 minus 15 14526119890 minus 15

1198902 15510119890 minus 15 23280119890 minus 15 minus48804119890 minus 16 minus14289119890 minus 15 13702119890 minus 15 22549119890 minus 15

1198903 0 0 0 0 0 0119904119902 41948119890 minus 15 27057119890 minus 15 26986119890 minus 15 24428119890 minus 15 18018119890 minus 15 26823119890 minus 15

Layingpoint

z

yG

x

Surfaceship

Transponder

Seabed

Figure 2 The calibration geometry for calibration of the transpon-der position

single transponder UTP aided inertial navigation was carriedout in 2003 as described in [33]

For ranging techniques like UTP to work the geo-graphical location of the transponders must be known Thepreferred method is to measure the position directly usingUSBL on a surface ship When the transponder deploymentis completed a surface ship with USBL and GPS sails aroundthe transponder in a circular motion collecting surface shipposition and distance information The calibration geometryis illustrated in Figure 2

In order to set the design framework let 119866 denote theglobal coordinate frame and let 119861 denote a coordinateframe attached to the vehicle usually denominated as body-fixed coordinate frameThe frame 119874 is the transceiver coor-dinates The position of transponder in the global coordinateframe is given by

119866119883119879=119866119883119863minus

119866

119861119877119861119883119863+119866

119861119877(

119861

119874119877119874119883119879+119861119883119874) (25)

where 119866119883119863is the position of the GPS in global coordinates

119861119883119863is the position of the GPS in vehicle coordinates 119874119883

119879

is the position of the transponder in transceiver coordinates119861119883119874is the position of the transceiver in vehicle coordinates

119866

119861119877is the rotation matrix from 119861 to 119866 and 119861

119874119877is the

rotation matrix from 119874 to 119861After the data collection we can get the distance between

the transponder and transceiver and the corresponding GPSposition in the global coordinates The attitude of the vehiclecan be measured by the heading sensors and the pitchrollsensor so 119866

119861119877is known We also know 119861119883

119863 119861119883119874 and 119861

119874119877

According to (25) we have

119874119883119879=

119861

119874119877minus1(

119866

119861119877minus1(119866119883119879minus119866119883119863) +119861119883119863minus119861119883119874) (26)

We denote 119874119883119879119894= [119874119909119879119894

119874119910119879119894

119874119911119879119894]

119879

(119894 = 1 2 119873)

as the position of the transponder in the transceiver coor-dinates in the 119894th measurement point The cost functionbecomes

119865 =

119873

sum

119894=1

(radic119874

1199092

119879119894+

119874

1199102

119879119894+

119874

1199112

119879119894minus 119871119894)

2

119873

(27)

Particles optimization algorithms have been introducedDefining each particle as a coordinate of the transponderTheparameters of particle swarmoptimization for simulationare the same as in Section 4 And a surface ship moves ina circular motion with a radius of 100 the real coordinatesof transponder are 119866119883

119903119879= [0 0 100]

119879 To simplify theproblem 119866

119861119877is unit matrix and 119861119883

119863minus119861119883119874= [0 0 0]

119879and ignore the sensor measurement error The simulationdata is shown in Table 3 Obviously the particle swarmalgorithm can search for the transponder coordinates andobtain accurate results At the same time CPSO shows thebest performance

Remark 2 One has

1198901 =119866119909119879minus119866119909119903119879 1198902 =

119866119910119879minus119866119910119903119879

1198903 =119866119911119879minus119866119911119903119879 119904119902 = radic1198901

2+ 11989022+ 11989032

(28)

7 Conclusion

In this paper a novel strategy to improve the performance ofparticle swarm optimization is proposed to apply in calibra-tion of the underwater transponder coordinates To improvethe population-based search optimization algorithm eachparticle is evolved along two different directions to generatetwo homologous particles The cost functions of two homol-ogous particles are calculated to keep the optimal one andto eliminate the poor one Then the next generation particleis updated It is regarded as CPSO Ten classify benchmarkfunctions are introduced to reflect the effectiveness of theproposed algorithmThe simulation results demonstrate thatthe unimodal function CPSO algorithm is superior toBPSO LWPSO EPSO and TVAC on the searching accuracystability and convergence speed However considering themultimodal function the performance of TVAC is superiorto CPSO

Mathematical Problems in Engineering 11

Secondly to further improve the performance of CPSOthe ECPSO is proposed by combining the CPSO and theTVAC In the initial period of the evolution the individualexperience is a significant aspect with larger accelerationcoefficient and in the final period the swarm experience issuperior with a greater acceleration coefficient Simultane-ously the evolution for each particle at any time is towardstwo different inertia directions to generate two homologousparticles and to obtain its next generation particles Thesimulations show the effectiveness of multimodal function byusing ECPSOWith the incensement of benchmark functionsdimensions the accuracy and stability of each algorithm willdecrease but CPSO and EPSO display the best performance

At last the strategy to calibrate the underwater transpon-der coordinates using particle swarm algorithm is intro-duced As the cost function for transponder coordinates is aunimodal function CPSO shows better performance than theother algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the Natural Science Foun-dation of China (51179038 1109043 51309067E091002) andthe Program of New Century Excellent Talents in University(NCET-10-0053)

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the 1995 IEEE International Conference on NeuralNetworks vol 4 no 2 pp 1942ndash1948 December 1995

[2] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[3] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[4] M S Arumugam M V C Rao and A W C Tan ldquoA noveland effective particle swarm optimization like algorithm withextrapolation techniquerdquo Applied Soft Computing Journal vol9 no 1 pp 308ndash320 2009

[5] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 1999

[6] G Chen X Huang J Jia et al ldquoNatural exponential inertiaweight strategy in particle swarm optimizationrdquo in Proceedingsof the 6thWorld Congress on Intelligent Control and Automation(WCICA rsquo06) vol 1 pp 3672ndash3675 IEEE June 2006

[7] Q Ni and J Deng ldquoA new logistic dynamic particle swarm opti-mization algorithm based on random topologyrdquo The ScientificWorld Journal vol 2013 Article ID 409167 8 pages 2013

[8] M M Noel ldquoA new gradient based particle swarm optimiza-tion algorithm for accurate computation of global minimumrdquoApplied Soft Computing Journal vol 12 no 1 pp 353ndash359 2012

[9] M G Epitropakis V P Plagianakos and M N VrahatisldquoEvolving cognitive and social experience in particle swarmoptimization through differential evolutionrdquo in Proceedings ofthe IEEE Congress on Evolutionary Computation (CEC rsquo10) pp1ndash8 IEEE July 2010

[10] A A Mousa M A El-Shorbagy and W F Abd-El-WahedldquoLocal search based hybrid particle swarm optimization algo-rithm for multiobjective optimizationrdquo Swarm and Evolution-ary Computation vol 3 pp 1ndash14 2012

[11] M H Moradi and M Abedini ldquoA combination of genetic algo-rithm and particle swarm optimization for optimal DG locationand sizing in distribution systemsrdquo International Journal ofElectrical Power and Energy Systems vol 34 no 1 pp 66ndash742012

[12] W D Chang J P Cheng M C Hsu et al ldquoParameteridentification of nonlinear systems using a particle swarmoptimization approachrdquo in Proceedings of the 3rd InternationalConference on Networking and Computing (ICNC rsquo12) pp 113ndash117 IEEE 2012

[13] J J Soon and K S Low ldquoPhotovoltaic model identificationusing particle swarm optimization with inverse barrier con-straintrdquo IEEE Transactions on Power Electronics vol 27 no 9pp 3975ndash3983 2012

[14] B Jiang N Wang and L Wang ldquoParameter identification forsolid oxide fuel cells using cooperative barebone particle swarmoptimization with hybrid learningrdquo International Journal ofHydrogen Energy vol 39 no 1 pp 532ndash542 2013

[15] A Alfi ldquoParticle swarm optimization algorithm with DynamicInertia Weight for online parameter identification applied toLorenz chaotic systemrdquo International Journal of InnovativeComputing Information and Control vol 8 no 2 pp 1191ndash12042012

[16] X Hu S Shi and X Gu ldquoAn improved particle swarm opti-mization algorithm for wireless sensor networks localizationrdquoin Proceedings of the 8th International Conference on WirelessCommunications Networking and Mobile Computing (WiCOMrsquo12) pp 1ndash4 Shanghai China September 2012

[17] F Sun J Yu and D Xu ldquoVisual measurement and control forunderwater robots a surveyrdquo in Proceedings of the 25th ChineseControl andDecision Conference (CCDC rsquo13) pp 333ndash338 IEEE2013

[18] V Djapic D Nad G Ferri et al ldquoNovel method for underwaternavigation aiding using a companion underwater robot asa guiding platformsrdquo in Proceedings of the 2013 MTSIEEEOCEANS-Bergen pp 1ndash10 IEEE June 2013

[19] A Caiti V Calabro D Meucci et al ldquoUnderwater Robots PastPresent and Futurerdquo 422ndash437

[20] P Krishnamurthy and F Khorrami ldquoA self-aligning underwaternavigation system based on fusion ofmultiple sensors includingDVL and IMUrdquo in Proceedings of the 9th Asian Control Confer-ence (ASCC rsquo13) pp 1ndash6 IEEE 2013

[21] S Wang and F Kang ldquoResearch and design of UUV navigationand control integrative simulation system based on compo-nentrdquo Intelligent Information Management vol 4 no 5 pp 181ndash187 2012

[22] T Kashima A Asada and T Ura ldquoThe positioning sys-tem integrated LBL and SSBL using seafloor acoustic mirrortransponderrdquo in Proceedings of the IEEE International Underwa-ter Technology Symposium (UT rsquo13) March 2013

12 Mathematical Problems in Engineering

[23] P Batista C Silvestre and P Oliveira ldquoGAS tightly coupledLBLUSBL position and velocity filter for underwater vehiclesrdquoin Proceedings of the European Control Conference (ECC rsquo13) pp2982ndash2987 Zurich Switzerland 2013

[24] M Morgado P Oliveira and C Silvestre ldquoTightly coupledultrashort baseline and inertial navigation system for under-water vehicles an experimental validationrdquo Journal of FieldRobotics vol 30 no 1 pp 142ndash170 2013

[25] Y U Min and H Junyin ldquoThe calibration of the USBL trans-ducer array for Long-range precision underwater positioningrdquoin Proceedings of the IEEE 10th International Conference onSignal Processing (ICSP rsquo10) pp 2357ndash2360 IEEE October 2010

[26] D Ji Y Li and J Liu ldquoSeafloor transponder calibration usingimproved perpendiculars intersectionrdquoAppliedOceanResearchvol 32 no 3 pp 261ndash266 2010

[27] B Jiao Z Lian and Q Chen ldquoA dynamic global and local com-bined particle swarm optimization algorithmrdquo Chaos Solitonsand Fractals vol 42 no 5 pp 2688ndash2695 2009

[28] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 IEEE May 1998

[29] M Schwaab E C Biscaia Jr J L Monteiro and J CPinto ldquoNonlinear parameter estimation through particle swarmoptimizationrdquo Chemical Engineering Science vol 63 no 6 pp1542ndash1552 2008

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII pp591ndash600 Springer Berlin Germany 1998

[31] A Chander A Chatterjee and P Siarry ldquoA new social andmomentum component adaptive PSO algorithm for imagesegmentationrdquo Expert Systems with Applications vol 38 no 5pp 4998ndash5004 2011

[32] M Morgado P Batista P Oliveira and C Silvestre ldquoPositionUSBLDVL sensor-based navigation filter in the presence ofunknown ocean currentsrdquoAutomatica vol 47 no 12 pp 2604ndash2614 2011

[33] B Jalving K Gade O K Hagen and K Vestgard ldquoA toolboxof aiding techniques for the HUGIN AUV integrated inertialnavigation systemrdquo in Proceedings of the OCEANS 2003 vol 2pp 1146ndash1153 IEEE September 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 200 400 600 800 1000

0

005

01

015

02

025

03

035

Number of generations

Number of generations Number of generations

Number of generations

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

Mea

n be

st va

lue

0 200 400 600 800 1000

0

05

1

15

2

25

3

0 200 400 600 800 1000

0

05

1

15

2

25

0 200 400 600 800 1000

0

002

004

006

008

01

012

014

016

minus05

minus05

minus005

minus002

BPSOLWPSOEPSO

TVACCPSOECPSO

BPSOLWPSOEPSO

TVACCPSOECPSO

f8f7

f10f9

Figure 1 Variation of the average optimum value with time

the early part of the optimization and encourage the particlesto converge toward the global optima at the end of the searchComparing to TVAC andCPSO the proposed new algorithmECPSO is appropriate for multimodal function search

With the increase of benchmark functions dimensionthe searching accuracy and stability for each algorithmare decreased The performances of CPSO and ECPSO aresuperior to the other algorithms With regard to multimodalfunction the performances of ECPSOandTVACare superiorto CPSO The average fitness is varied as in Table 2 FromFigure 1 it shows that the order of searching speed fromhigh to low is BPSO (green solid curve) LWPSO (red circle)EPSO (blue triangle) TVAC (black dot curve) CPSO (bluedash-dot) and ECPSO (purples dash) It is obvious that theperformances of EPSO and CPSO are more effective than theother algorithms

6 Calibrate the UnderwaterTransponder Coordinates

After two decades of dedicated research and developmentunmanned underwater vehicles (UUV) have been acceptedby an increasing number of users in bothmilitary and civilianinstitutions The design and implementation of navigationsystems stand out as one of the most critical steps towardsthe successful operation of autonomous vehicles The qualityof the overall estimates of the navigation system dramaticallyinfluences the capability of the vehicles to perform precision-demanding tasks [32] From a navigation point of view asingle range transponder may be regarded as an underwaterlighthouse providingUUVwith the ranges relative to its fixedgeographical location Single transponder navigation is not anew concept As mentioned the first at-sea demonstration of

10 Mathematical Problems in Engineering

Table 3 The calibration of the underwater transponder coordinates

BPSO LWPSO EPSO TVAC CPSO ECPSO1198901 35371119890 minus 15 minus23573119890 minus 15 26542119890 minus 15 minus19813119890 minus 15 minus11700119890 minus 15 14526119890 minus 15

1198902 15510119890 minus 15 23280119890 minus 15 minus48804119890 minus 16 minus14289119890 minus 15 13702119890 minus 15 22549119890 minus 15

1198903 0 0 0 0 0 0119904119902 41948119890 minus 15 27057119890 minus 15 26986119890 minus 15 24428119890 minus 15 18018119890 minus 15 26823119890 minus 15

Layingpoint

z

yG

x

Surfaceship

Transponder

Seabed

Figure 2 The calibration geometry for calibration of the transpon-der position

single transponder UTP aided inertial navigation was carriedout in 2003 as described in [33]

For ranging techniques like UTP to work the geo-graphical location of the transponders must be known Thepreferred method is to measure the position directly usingUSBL on a surface ship When the transponder deploymentis completed a surface ship with USBL and GPS sails aroundthe transponder in a circular motion collecting surface shipposition and distance information The calibration geometryis illustrated in Figure 2

In order to set the design framework let 119866 denote theglobal coordinate frame and let 119861 denote a coordinateframe attached to the vehicle usually denominated as body-fixed coordinate frameThe frame 119874 is the transceiver coor-dinates The position of transponder in the global coordinateframe is given by

119866119883119879=119866119883119863minus

119866

119861119877119861119883119863+119866

119861119877(

119861

119874119877119874119883119879+119861119883119874) (25)

where 119866119883119863is the position of the GPS in global coordinates

119861119883119863is the position of the GPS in vehicle coordinates 119874119883

119879

is the position of the transponder in transceiver coordinates119861119883119874is the position of the transceiver in vehicle coordinates

119866

119861119877is the rotation matrix from 119861 to 119866 and 119861

119874119877is the

rotation matrix from 119874 to 119861After the data collection we can get the distance between

the transponder and transceiver and the corresponding GPSposition in the global coordinates The attitude of the vehiclecan be measured by the heading sensors and the pitchrollsensor so 119866

119861119877is known We also know 119861119883

119863 119861119883119874 and 119861

119874119877

According to (25) we have

119874119883119879=

119861

119874119877minus1(

119866

119861119877minus1(119866119883119879minus119866119883119863) +119861119883119863minus119861119883119874) (26)

We denote 119874119883119879119894= [119874119909119879119894

119874119910119879119894

119874119911119879119894]

119879

(119894 = 1 2 119873)

as the position of the transponder in the transceiver coor-dinates in the 119894th measurement point The cost functionbecomes

119865 =

119873

sum

119894=1

(radic119874

1199092

119879119894+

119874

1199102

119879119894+

119874

1199112

119879119894minus 119871119894)

2

119873

(27)

Particles optimization algorithms have been introducedDefining each particle as a coordinate of the transponderTheparameters of particle swarmoptimization for simulationare the same as in Section 4 And a surface ship moves ina circular motion with a radius of 100 the real coordinatesof transponder are 119866119883

119903119879= [0 0 100]

119879 To simplify theproblem 119866

119861119877is unit matrix and 119861119883

119863minus119861119883119874= [0 0 0]

119879and ignore the sensor measurement error The simulationdata is shown in Table 3 Obviously the particle swarmalgorithm can search for the transponder coordinates andobtain accurate results At the same time CPSO shows thebest performance

Remark 2 One has

1198901 =119866119909119879minus119866119909119903119879 1198902 =

119866119910119879minus119866119910119903119879

1198903 =119866119911119879minus119866119911119903119879 119904119902 = radic1198901

2+ 11989022+ 11989032

(28)

7 Conclusion

In this paper a novel strategy to improve the performance ofparticle swarm optimization is proposed to apply in calibra-tion of the underwater transponder coordinates To improvethe population-based search optimization algorithm eachparticle is evolved along two different directions to generatetwo homologous particles The cost functions of two homol-ogous particles are calculated to keep the optimal one andto eliminate the poor one Then the next generation particleis updated It is regarded as CPSO Ten classify benchmarkfunctions are introduced to reflect the effectiveness of theproposed algorithmThe simulation results demonstrate thatthe unimodal function CPSO algorithm is superior toBPSO LWPSO EPSO and TVAC on the searching accuracystability and convergence speed However considering themultimodal function the performance of TVAC is superiorto CPSO

Mathematical Problems in Engineering 11

Secondly to further improve the performance of CPSOthe ECPSO is proposed by combining the CPSO and theTVAC In the initial period of the evolution the individualexperience is a significant aspect with larger accelerationcoefficient and in the final period the swarm experience issuperior with a greater acceleration coefficient Simultane-ously the evolution for each particle at any time is towardstwo different inertia directions to generate two homologousparticles and to obtain its next generation particles Thesimulations show the effectiveness of multimodal function byusing ECPSOWith the incensement of benchmark functionsdimensions the accuracy and stability of each algorithm willdecrease but CPSO and EPSO display the best performance

At last the strategy to calibrate the underwater transpon-der coordinates using particle swarm algorithm is intro-duced As the cost function for transponder coordinates is aunimodal function CPSO shows better performance than theother algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the Natural Science Foun-dation of China (51179038 1109043 51309067E091002) andthe Program of New Century Excellent Talents in University(NCET-10-0053)

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the 1995 IEEE International Conference on NeuralNetworks vol 4 no 2 pp 1942ndash1948 December 1995

[2] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[3] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[4] M S Arumugam M V C Rao and A W C Tan ldquoA noveland effective particle swarm optimization like algorithm withextrapolation techniquerdquo Applied Soft Computing Journal vol9 no 1 pp 308ndash320 2009

[5] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 1999

[6] G Chen X Huang J Jia et al ldquoNatural exponential inertiaweight strategy in particle swarm optimizationrdquo in Proceedingsof the 6thWorld Congress on Intelligent Control and Automation(WCICA rsquo06) vol 1 pp 3672ndash3675 IEEE June 2006

[7] Q Ni and J Deng ldquoA new logistic dynamic particle swarm opti-mization algorithm based on random topologyrdquo The ScientificWorld Journal vol 2013 Article ID 409167 8 pages 2013

[8] M M Noel ldquoA new gradient based particle swarm optimiza-tion algorithm for accurate computation of global minimumrdquoApplied Soft Computing Journal vol 12 no 1 pp 353ndash359 2012

[9] M G Epitropakis V P Plagianakos and M N VrahatisldquoEvolving cognitive and social experience in particle swarmoptimization through differential evolutionrdquo in Proceedings ofthe IEEE Congress on Evolutionary Computation (CEC rsquo10) pp1ndash8 IEEE July 2010

[10] A A Mousa M A El-Shorbagy and W F Abd-El-WahedldquoLocal search based hybrid particle swarm optimization algo-rithm for multiobjective optimizationrdquo Swarm and Evolution-ary Computation vol 3 pp 1ndash14 2012

[11] M H Moradi and M Abedini ldquoA combination of genetic algo-rithm and particle swarm optimization for optimal DG locationand sizing in distribution systemsrdquo International Journal ofElectrical Power and Energy Systems vol 34 no 1 pp 66ndash742012

[12] W D Chang J P Cheng M C Hsu et al ldquoParameteridentification of nonlinear systems using a particle swarmoptimization approachrdquo in Proceedings of the 3rd InternationalConference on Networking and Computing (ICNC rsquo12) pp 113ndash117 IEEE 2012

[13] J J Soon and K S Low ldquoPhotovoltaic model identificationusing particle swarm optimization with inverse barrier con-straintrdquo IEEE Transactions on Power Electronics vol 27 no 9pp 3975ndash3983 2012

[14] B Jiang N Wang and L Wang ldquoParameter identification forsolid oxide fuel cells using cooperative barebone particle swarmoptimization with hybrid learningrdquo International Journal ofHydrogen Energy vol 39 no 1 pp 532ndash542 2013

[15] A Alfi ldquoParticle swarm optimization algorithm with DynamicInertia Weight for online parameter identification applied toLorenz chaotic systemrdquo International Journal of InnovativeComputing Information and Control vol 8 no 2 pp 1191ndash12042012

[16] X Hu S Shi and X Gu ldquoAn improved particle swarm opti-mization algorithm for wireless sensor networks localizationrdquoin Proceedings of the 8th International Conference on WirelessCommunications Networking and Mobile Computing (WiCOMrsquo12) pp 1ndash4 Shanghai China September 2012

[17] F Sun J Yu and D Xu ldquoVisual measurement and control forunderwater robots a surveyrdquo in Proceedings of the 25th ChineseControl andDecision Conference (CCDC rsquo13) pp 333ndash338 IEEE2013

[18] V Djapic D Nad G Ferri et al ldquoNovel method for underwaternavigation aiding using a companion underwater robot asa guiding platformsrdquo in Proceedings of the 2013 MTSIEEEOCEANS-Bergen pp 1ndash10 IEEE June 2013

[19] A Caiti V Calabro D Meucci et al ldquoUnderwater Robots PastPresent and Futurerdquo 422ndash437

[20] P Krishnamurthy and F Khorrami ldquoA self-aligning underwaternavigation system based on fusion ofmultiple sensors includingDVL and IMUrdquo in Proceedings of the 9th Asian Control Confer-ence (ASCC rsquo13) pp 1ndash6 IEEE 2013

[21] S Wang and F Kang ldquoResearch and design of UUV navigationand control integrative simulation system based on compo-nentrdquo Intelligent Information Management vol 4 no 5 pp 181ndash187 2012

[22] T Kashima A Asada and T Ura ldquoThe positioning sys-tem integrated LBL and SSBL using seafloor acoustic mirrortransponderrdquo in Proceedings of the IEEE International Underwa-ter Technology Symposium (UT rsquo13) March 2013

12 Mathematical Problems in Engineering

[23] P Batista C Silvestre and P Oliveira ldquoGAS tightly coupledLBLUSBL position and velocity filter for underwater vehiclesrdquoin Proceedings of the European Control Conference (ECC rsquo13) pp2982ndash2987 Zurich Switzerland 2013

[24] M Morgado P Oliveira and C Silvestre ldquoTightly coupledultrashort baseline and inertial navigation system for under-water vehicles an experimental validationrdquo Journal of FieldRobotics vol 30 no 1 pp 142ndash170 2013

[25] Y U Min and H Junyin ldquoThe calibration of the USBL trans-ducer array for Long-range precision underwater positioningrdquoin Proceedings of the IEEE 10th International Conference onSignal Processing (ICSP rsquo10) pp 2357ndash2360 IEEE October 2010

[26] D Ji Y Li and J Liu ldquoSeafloor transponder calibration usingimproved perpendiculars intersectionrdquoAppliedOceanResearchvol 32 no 3 pp 261ndash266 2010

[27] B Jiao Z Lian and Q Chen ldquoA dynamic global and local com-bined particle swarm optimization algorithmrdquo Chaos Solitonsand Fractals vol 42 no 5 pp 2688ndash2695 2009

[28] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 IEEE May 1998

[29] M Schwaab E C Biscaia Jr J L Monteiro and J CPinto ldquoNonlinear parameter estimation through particle swarmoptimizationrdquo Chemical Engineering Science vol 63 no 6 pp1542ndash1552 2008

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII pp591ndash600 Springer Berlin Germany 1998

[31] A Chander A Chatterjee and P Siarry ldquoA new social andmomentum component adaptive PSO algorithm for imagesegmentationrdquo Expert Systems with Applications vol 38 no 5pp 4998ndash5004 2011

[32] M Morgado P Batista P Oliveira and C Silvestre ldquoPositionUSBLDVL sensor-based navigation filter in the presence ofunknown ocean currentsrdquoAutomatica vol 47 no 12 pp 2604ndash2614 2011

[33] B Jalving K Gade O K Hagen and K Vestgard ldquoA toolboxof aiding techniques for the HUGIN AUV integrated inertialnavigation systemrdquo in Proceedings of the OCEANS 2003 vol 2pp 1146ndash1153 IEEE September 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 3 The calibration of the underwater transponder coordinates

BPSO LWPSO EPSO TVAC CPSO ECPSO1198901 35371119890 minus 15 minus23573119890 minus 15 26542119890 minus 15 minus19813119890 minus 15 minus11700119890 minus 15 14526119890 minus 15

1198902 15510119890 minus 15 23280119890 minus 15 minus48804119890 minus 16 minus14289119890 minus 15 13702119890 minus 15 22549119890 minus 15

1198903 0 0 0 0 0 0119904119902 41948119890 minus 15 27057119890 minus 15 26986119890 minus 15 24428119890 minus 15 18018119890 minus 15 26823119890 minus 15

Layingpoint

z

yG

x

Surfaceship

Transponder

Seabed

Figure 2 The calibration geometry for calibration of the transpon-der position

single transponder UTP aided inertial navigation was carriedout in 2003 as described in [33]

For ranging techniques like UTP to work the geo-graphical location of the transponders must be known Thepreferred method is to measure the position directly usingUSBL on a surface ship When the transponder deploymentis completed a surface ship with USBL and GPS sails aroundthe transponder in a circular motion collecting surface shipposition and distance information The calibration geometryis illustrated in Figure 2

In order to set the design framework let 119866 denote theglobal coordinate frame and let 119861 denote a coordinateframe attached to the vehicle usually denominated as body-fixed coordinate frameThe frame 119874 is the transceiver coor-dinates The position of transponder in the global coordinateframe is given by

119866119883119879=119866119883119863minus

119866

119861119877119861119883119863+119866

119861119877(

119861

119874119877119874119883119879+119861119883119874) (25)

where 119866119883119863is the position of the GPS in global coordinates

119861119883119863is the position of the GPS in vehicle coordinates 119874119883

119879

is the position of the transponder in transceiver coordinates119861119883119874is the position of the transceiver in vehicle coordinates

119866

119861119877is the rotation matrix from 119861 to 119866 and 119861

119874119877is the

rotation matrix from 119874 to 119861After the data collection we can get the distance between

the transponder and transceiver and the corresponding GPSposition in the global coordinates The attitude of the vehiclecan be measured by the heading sensors and the pitchrollsensor so 119866

119861119877is known We also know 119861119883

119863 119861119883119874 and 119861

119874119877

According to (25) we have

119874119883119879=

119861

119874119877minus1(

119866

119861119877minus1(119866119883119879minus119866119883119863) +119861119883119863minus119861119883119874) (26)

We denote 119874119883119879119894= [119874119909119879119894

119874119910119879119894

119874119911119879119894]

119879

(119894 = 1 2 119873)

as the position of the transponder in the transceiver coor-dinates in the 119894th measurement point The cost functionbecomes

119865 =

119873

sum

119894=1

(radic119874

1199092

119879119894+

119874

1199102

119879119894+

119874

1199112

119879119894minus 119871119894)

2

119873

(27)

Particles optimization algorithms have been introducedDefining each particle as a coordinate of the transponderTheparameters of particle swarmoptimization for simulationare the same as in Section 4 And a surface ship moves ina circular motion with a radius of 100 the real coordinatesof transponder are 119866119883

119903119879= [0 0 100]

119879 To simplify theproblem 119866

119861119877is unit matrix and 119861119883

119863minus119861119883119874= [0 0 0]

119879and ignore the sensor measurement error The simulationdata is shown in Table 3 Obviously the particle swarmalgorithm can search for the transponder coordinates andobtain accurate results At the same time CPSO shows thebest performance

Remark 2 One has

1198901 =119866119909119879minus119866119909119903119879 1198902 =

119866119910119879minus119866119910119903119879

1198903 =119866119911119879minus119866119911119903119879 119904119902 = radic1198901

2+ 11989022+ 11989032

(28)

7 Conclusion

In this paper a novel strategy to improve the performance ofparticle swarm optimization is proposed to apply in calibra-tion of the underwater transponder coordinates To improvethe population-based search optimization algorithm eachparticle is evolved along two different directions to generatetwo homologous particles The cost functions of two homol-ogous particles are calculated to keep the optimal one andto eliminate the poor one Then the next generation particleis updated It is regarded as CPSO Ten classify benchmarkfunctions are introduced to reflect the effectiveness of theproposed algorithmThe simulation results demonstrate thatthe unimodal function CPSO algorithm is superior toBPSO LWPSO EPSO and TVAC on the searching accuracystability and convergence speed However considering themultimodal function the performance of TVAC is superiorto CPSO

Mathematical Problems in Engineering 11

Secondly to further improve the performance of CPSOthe ECPSO is proposed by combining the CPSO and theTVAC In the initial period of the evolution the individualexperience is a significant aspect with larger accelerationcoefficient and in the final period the swarm experience issuperior with a greater acceleration coefficient Simultane-ously the evolution for each particle at any time is towardstwo different inertia directions to generate two homologousparticles and to obtain its next generation particles Thesimulations show the effectiveness of multimodal function byusing ECPSOWith the incensement of benchmark functionsdimensions the accuracy and stability of each algorithm willdecrease but CPSO and EPSO display the best performance

At last the strategy to calibrate the underwater transpon-der coordinates using particle swarm algorithm is intro-duced As the cost function for transponder coordinates is aunimodal function CPSO shows better performance than theother algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the Natural Science Foun-dation of China (51179038 1109043 51309067E091002) andthe Program of New Century Excellent Talents in University(NCET-10-0053)

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the 1995 IEEE International Conference on NeuralNetworks vol 4 no 2 pp 1942ndash1948 December 1995

[2] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[3] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[4] M S Arumugam M V C Rao and A W C Tan ldquoA noveland effective particle swarm optimization like algorithm withextrapolation techniquerdquo Applied Soft Computing Journal vol9 no 1 pp 308ndash320 2009

[5] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 1999

[6] G Chen X Huang J Jia et al ldquoNatural exponential inertiaweight strategy in particle swarm optimizationrdquo in Proceedingsof the 6thWorld Congress on Intelligent Control and Automation(WCICA rsquo06) vol 1 pp 3672ndash3675 IEEE June 2006

[7] Q Ni and J Deng ldquoA new logistic dynamic particle swarm opti-mization algorithm based on random topologyrdquo The ScientificWorld Journal vol 2013 Article ID 409167 8 pages 2013

[8] M M Noel ldquoA new gradient based particle swarm optimiza-tion algorithm for accurate computation of global minimumrdquoApplied Soft Computing Journal vol 12 no 1 pp 353ndash359 2012

[9] M G Epitropakis V P Plagianakos and M N VrahatisldquoEvolving cognitive and social experience in particle swarmoptimization through differential evolutionrdquo in Proceedings ofthe IEEE Congress on Evolutionary Computation (CEC rsquo10) pp1ndash8 IEEE July 2010

[10] A A Mousa M A El-Shorbagy and W F Abd-El-WahedldquoLocal search based hybrid particle swarm optimization algo-rithm for multiobjective optimizationrdquo Swarm and Evolution-ary Computation vol 3 pp 1ndash14 2012

[11] M H Moradi and M Abedini ldquoA combination of genetic algo-rithm and particle swarm optimization for optimal DG locationand sizing in distribution systemsrdquo International Journal ofElectrical Power and Energy Systems vol 34 no 1 pp 66ndash742012

[12] W D Chang J P Cheng M C Hsu et al ldquoParameteridentification of nonlinear systems using a particle swarmoptimization approachrdquo in Proceedings of the 3rd InternationalConference on Networking and Computing (ICNC rsquo12) pp 113ndash117 IEEE 2012

[13] J J Soon and K S Low ldquoPhotovoltaic model identificationusing particle swarm optimization with inverse barrier con-straintrdquo IEEE Transactions on Power Electronics vol 27 no 9pp 3975ndash3983 2012

[14] B Jiang N Wang and L Wang ldquoParameter identification forsolid oxide fuel cells using cooperative barebone particle swarmoptimization with hybrid learningrdquo International Journal ofHydrogen Energy vol 39 no 1 pp 532ndash542 2013

[15] A Alfi ldquoParticle swarm optimization algorithm with DynamicInertia Weight for online parameter identification applied toLorenz chaotic systemrdquo International Journal of InnovativeComputing Information and Control vol 8 no 2 pp 1191ndash12042012

[16] X Hu S Shi and X Gu ldquoAn improved particle swarm opti-mization algorithm for wireless sensor networks localizationrdquoin Proceedings of the 8th International Conference on WirelessCommunications Networking and Mobile Computing (WiCOMrsquo12) pp 1ndash4 Shanghai China September 2012

[17] F Sun J Yu and D Xu ldquoVisual measurement and control forunderwater robots a surveyrdquo in Proceedings of the 25th ChineseControl andDecision Conference (CCDC rsquo13) pp 333ndash338 IEEE2013

[18] V Djapic D Nad G Ferri et al ldquoNovel method for underwaternavigation aiding using a companion underwater robot asa guiding platformsrdquo in Proceedings of the 2013 MTSIEEEOCEANS-Bergen pp 1ndash10 IEEE June 2013

[19] A Caiti V Calabro D Meucci et al ldquoUnderwater Robots PastPresent and Futurerdquo 422ndash437

[20] P Krishnamurthy and F Khorrami ldquoA self-aligning underwaternavigation system based on fusion ofmultiple sensors includingDVL and IMUrdquo in Proceedings of the 9th Asian Control Confer-ence (ASCC rsquo13) pp 1ndash6 IEEE 2013

[21] S Wang and F Kang ldquoResearch and design of UUV navigationand control integrative simulation system based on compo-nentrdquo Intelligent Information Management vol 4 no 5 pp 181ndash187 2012

[22] T Kashima A Asada and T Ura ldquoThe positioning sys-tem integrated LBL and SSBL using seafloor acoustic mirrortransponderrdquo in Proceedings of the IEEE International Underwa-ter Technology Symposium (UT rsquo13) March 2013

12 Mathematical Problems in Engineering

[23] P Batista C Silvestre and P Oliveira ldquoGAS tightly coupledLBLUSBL position and velocity filter for underwater vehiclesrdquoin Proceedings of the European Control Conference (ECC rsquo13) pp2982ndash2987 Zurich Switzerland 2013

[24] M Morgado P Oliveira and C Silvestre ldquoTightly coupledultrashort baseline and inertial navigation system for under-water vehicles an experimental validationrdquo Journal of FieldRobotics vol 30 no 1 pp 142ndash170 2013

[25] Y U Min and H Junyin ldquoThe calibration of the USBL trans-ducer array for Long-range precision underwater positioningrdquoin Proceedings of the IEEE 10th International Conference onSignal Processing (ICSP rsquo10) pp 2357ndash2360 IEEE October 2010

[26] D Ji Y Li and J Liu ldquoSeafloor transponder calibration usingimproved perpendiculars intersectionrdquoAppliedOceanResearchvol 32 no 3 pp 261ndash266 2010

[27] B Jiao Z Lian and Q Chen ldquoA dynamic global and local com-bined particle swarm optimization algorithmrdquo Chaos Solitonsand Fractals vol 42 no 5 pp 2688ndash2695 2009

[28] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 IEEE May 1998

[29] M Schwaab E C Biscaia Jr J L Monteiro and J CPinto ldquoNonlinear parameter estimation through particle swarmoptimizationrdquo Chemical Engineering Science vol 63 no 6 pp1542ndash1552 2008

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII pp591ndash600 Springer Berlin Germany 1998

[31] A Chander A Chatterjee and P Siarry ldquoA new social andmomentum component adaptive PSO algorithm for imagesegmentationrdquo Expert Systems with Applications vol 38 no 5pp 4998ndash5004 2011

[32] M Morgado P Batista P Oliveira and C Silvestre ldquoPositionUSBLDVL sensor-based navigation filter in the presence ofunknown ocean currentsrdquoAutomatica vol 47 no 12 pp 2604ndash2614 2011

[33] B Jalving K Gade O K Hagen and K Vestgard ldquoA toolboxof aiding techniques for the HUGIN AUV integrated inertialnavigation systemrdquo in Proceedings of the OCEANS 2003 vol 2pp 1146ndash1153 IEEE September 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Secondly to further improve the performance of CPSOthe ECPSO is proposed by combining the CPSO and theTVAC In the initial period of the evolution the individualexperience is a significant aspect with larger accelerationcoefficient and in the final period the swarm experience issuperior with a greater acceleration coefficient Simultane-ously the evolution for each particle at any time is towardstwo different inertia directions to generate two homologousparticles and to obtain its next generation particles Thesimulations show the effectiveness of multimodal function byusing ECPSOWith the incensement of benchmark functionsdimensions the accuracy and stability of each algorithm willdecrease but CPSO and EPSO display the best performance

At last the strategy to calibrate the underwater transpon-der coordinates using particle swarm algorithm is intro-duced As the cost function for transponder coordinates is aunimodal function CPSO shows better performance than theother algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the Natural Science Foun-dation of China (51179038 1109043 51309067E091002) andthe Program of New Century Excellent Talents in University(NCET-10-0053)

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the 1995 IEEE International Conference on NeuralNetworks vol 4 no 2 pp 1942ndash1948 December 1995

[2] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[3] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[4] M S Arumugam M V C Rao and A W C Tan ldquoA noveland effective particle swarm optimization like algorithm withextrapolation techniquerdquo Applied Soft Computing Journal vol9 no 1 pp 308ndash320 2009

[5] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 1999

[6] G Chen X Huang J Jia et al ldquoNatural exponential inertiaweight strategy in particle swarm optimizationrdquo in Proceedingsof the 6thWorld Congress on Intelligent Control and Automation(WCICA rsquo06) vol 1 pp 3672ndash3675 IEEE June 2006

[7] Q Ni and J Deng ldquoA new logistic dynamic particle swarm opti-mization algorithm based on random topologyrdquo The ScientificWorld Journal vol 2013 Article ID 409167 8 pages 2013

[8] M M Noel ldquoA new gradient based particle swarm optimiza-tion algorithm for accurate computation of global minimumrdquoApplied Soft Computing Journal vol 12 no 1 pp 353ndash359 2012

[9] M G Epitropakis V P Plagianakos and M N VrahatisldquoEvolving cognitive and social experience in particle swarmoptimization through differential evolutionrdquo in Proceedings ofthe IEEE Congress on Evolutionary Computation (CEC rsquo10) pp1ndash8 IEEE July 2010

[10] A A Mousa M A El-Shorbagy and W F Abd-El-WahedldquoLocal search based hybrid particle swarm optimization algo-rithm for multiobjective optimizationrdquo Swarm and Evolution-ary Computation vol 3 pp 1ndash14 2012

[11] M H Moradi and M Abedini ldquoA combination of genetic algo-rithm and particle swarm optimization for optimal DG locationand sizing in distribution systemsrdquo International Journal ofElectrical Power and Energy Systems vol 34 no 1 pp 66ndash742012

[12] W D Chang J P Cheng M C Hsu et al ldquoParameteridentification of nonlinear systems using a particle swarmoptimization approachrdquo in Proceedings of the 3rd InternationalConference on Networking and Computing (ICNC rsquo12) pp 113ndash117 IEEE 2012

[13] J J Soon and K S Low ldquoPhotovoltaic model identificationusing particle swarm optimization with inverse barrier con-straintrdquo IEEE Transactions on Power Electronics vol 27 no 9pp 3975ndash3983 2012

[14] B Jiang N Wang and L Wang ldquoParameter identification forsolid oxide fuel cells using cooperative barebone particle swarmoptimization with hybrid learningrdquo International Journal ofHydrogen Energy vol 39 no 1 pp 532ndash542 2013

[15] A Alfi ldquoParticle swarm optimization algorithm with DynamicInertia Weight for online parameter identification applied toLorenz chaotic systemrdquo International Journal of InnovativeComputing Information and Control vol 8 no 2 pp 1191ndash12042012

[16] X Hu S Shi and X Gu ldquoAn improved particle swarm opti-mization algorithm for wireless sensor networks localizationrdquoin Proceedings of the 8th International Conference on WirelessCommunications Networking and Mobile Computing (WiCOMrsquo12) pp 1ndash4 Shanghai China September 2012

[17] F Sun J Yu and D Xu ldquoVisual measurement and control forunderwater robots a surveyrdquo in Proceedings of the 25th ChineseControl andDecision Conference (CCDC rsquo13) pp 333ndash338 IEEE2013

[18] V Djapic D Nad G Ferri et al ldquoNovel method for underwaternavigation aiding using a companion underwater robot asa guiding platformsrdquo in Proceedings of the 2013 MTSIEEEOCEANS-Bergen pp 1ndash10 IEEE June 2013

[19] A Caiti V Calabro D Meucci et al ldquoUnderwater Robots PastPresent and Futurerdquo 422ndash437

[20] P Krishnamurthy and F Khorrami ldquoA self-aligning underwaternavigation system based on fusion ofmultiple sensors includingDVL and IMUrdquo in Proceedings of the 9th Asian Control Confer-ence (ASCC rsquo13) pp 1ndash6 IEEE 2013

[21] S Wang and F Kang ldquoResearch and design of UUV navigationand control integrative simulation system based on compo-nentrdquo Intelligent Information Management vol 4 no 5 pp 181ndash187 2012

[22] T Kashima A Asada and T Ura ldquoThe positioning sys-tem integrated LBL and SSBL using seafloor acoustic mirrortransponderrdquo in Proceedings of the IEEE International Underwa-ter Technology Symposium (UT rsquo13) March 2013

12 Mathematical Problems in Engineering

[23] P Batista C Silvestre and P Oliveira ldquoGAS tightly coupledLBLUSBL position and velocity filter for underwater vehiclesrdquoin Proceedings of the European Control Conference (ECC rsquo13) pp2982ndash2987 Zurich Switzerland 2013

[24] M Morgado P Oliveira and C Silvestre ldquoTightly coupledultrashort baseline and inertial navigation system for under-water vehicles an experimental validationrdquo Journal of FieldRobotics vol 30 no 1 pp 142ndash170 2013

[25] Y U Min and H Junyin ldquoThe calibration of the USBL trans-ducer array for Long-range precision underwater positioningrdquoin Proceedings of the IEEE 10th International Conference onSignal Processing (ICSP rsquo10) pp 2357ndash2360 IEEE October 2010

[26] D Ji Y Li and J Liu ldquoSeafloor transponder calibration usingimproved perpendiculars intersectionrdquoAppliedOceanResearchvol 32 no 3 pp 261ndash266 2010

[27] B Jiao Z Lian and Q Chen ldquoA dynamic global and local com-bined particle swarm optimization algorithmrdquo Chaos Solitonsand Fractals vol 42 no 5 pp 2688ndash2695 2009

[28] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 IEEE May 1998

[29] M Schwaab E C Biscaia Jr J L Monteiro and J CPinto ldquoNonlinear parameter estimation through particle swarmoptimizationrdquo Chemical Engineering Science vol 63 no 6 pp1542ndash1552 2008

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII pp591ndash600 Springer Berlin Germany 1998

[31] A Chander A Chatterjee and P Siarry ldquoA new social andmomentum component adaptive PSO algorithm for imagesegmentationrdquo Expert Systems with Applications vol 38 no 5pp 4998ndash5004 2011

[32] M Morgado P Batista P Oliveira and C Silvestre ldquoPositionUSBLDVL sensor-based navigation filter in the presence ofunknown ocean currentsrdquoAutomatica vol 47 no 12 pp 2604ndash2614 2011

[33] B Jalving K Gade O K Hagen and K Vestgard ldquoA toolboxof aiding techniques for the HUGIN AUV integrated inertialnavigation systemrdquo in Proceedings of the OCEANS 2003 vol 2pp 1146ndash1153 IEEE September 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[23] P Batista C Silvestre and P Oliveira ldquoGAS tightly coupledLBLUSBL position and velocity filter for underwater vehiclesrdquoin Proceedings of the European Control Conference (ECC rsquo13) pp2982ndash2987 Zurich Switzerland 2013

[24] M Morgado P Oliveira and C Silvestre ldquoTightly coupledultrashort baseline and inertial navigation system for under-water vehicles an experimental validationrdquo Journal of FieldRobotics vol 30 no 1 pp 142ndash170 2013

[25] Y U Min and H Junyin ldquoThe calibration of the USBL trans-ducer array for Long-range precision underwater positioningrdquoin Proceedings of the IEEE 10th International Conference onSignal Processing (ICSP rsquo10) pp 2357ndash2360 IEEE October 2010

[26] D Ji Y Li and J Liu ldquoSeafloor transponder calibration usingimproved perpendiculars intersectionrdquoAppliedOceanResearchvol 32 no 3 pp 261ndash266 2010

[27] B Jiao Z Lian and Q Chen ldquoA dynamic global and local com-bined particle swarm optimization algorithmrdquo Chaos Solitonsand Fractals vol 42 no 5 pp 2688ndash2695 2009

[28] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 IEEE May 1998

[29] M Schwaab E C Biscaia Jr J L Monteiro and J CPinto ldquoNonlinear parameter estimation through particle swarmoptimizationrdquo Chemical Engineering Science vol 63 no 6 pp1542ndash1552 2008

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII pp591ndash600 Springer Berlin Germany 1998

[31] A Chander A Chatterjee and P Siarry ldquoA new social andmomentum component adaptive PSO algorithm for imagesegmentationrdquo Expert Systems with Applications vol 38 no 5pp 4998ndash5004 2011

[32] M Morgado P Batista P Oliveira and C Silvestre ldquoPositionUSBLDVL sensor-based navigation filter in the presence ofunknown ocean currentsrdquoAutomatica vol 47 no 12 pp 2604ndash2614 2011

[33] B Jalving K Gade O K Hagen and K Vestgard ldquoA toolboxof aiding techniques for the HUGIN AUV integrated inertialnavigation systemrdquo in Proceedings of the OCEANS 2003 vol 2pp 1146ndash1153 IEEE September 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of