research article an optimization algorithm with novel rfa-pso … · 2019. 7. 31. · genetic...

Research ArticleAn Optimization Algorithm with Novel RFA-PSO CooperativeEvolution Applications to Parameter Decision of a Snake Robot

Qin Gao12 Zhelong Wang12 and Hongyi Li2

1School of Control Science and Engineering Dalian University of Technology Dalian 116024 China2State Key Laboratory of Robotics Shenyang Institute of Automation Chinese Academy of Sciences Shenyang 110000 China

Correspondence should be addressed to Qin Gao onlygaoqingmailcom

Received 14 July 2014 Revised 30 October 2014 Accepted 13 November 2014

Academic Editor Victor Santibanez

Copyright copy 2015 Qin Gao et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The success to design a hybrid optimization algorithm depends on how tomake full use of the effect of exploration and exploitationcarried by agents To improve the exploration and exploitation property of the agents we present a hybrid optimization algorithmwith both local and global search capabilities by combining the global search property of rain forest algorithm (RFA) and the rapidconvergence of PSO Originally two kinds of agents RFAAs and PSOAs are introduced to carry out exploration and exploitationrespectively In order to improve population diversification uniform distribution and adaptive range division are carried out byRFAAs in flexible scale during the iteration A further improvement has been provided to enhance the convergence rate andprocessing speed by combining PSO algorithm with potential guides found by both RFAAs and PSOAs Since several contingentlocal minima conditions may happen to PSO special agent transformation is suggested to provide information exchanging andcooperative coevolution between RFAAs and PSOAs Effectiveness and efficiency of the proposed algorithm are compared withseveral algorithms in the various benchmark function problems Finally engineering design optimization problems taken from thegait control of a snake-like robot are implemented successfully by the proposed RFA-PSO

1 Introduction

Genetic algorithm (GA) and particle swarm optimization(PSO) are currently being used in a variety of applicationswith great success [1ndash3] Their main advantage is to deal withcomplex nonconvex optimization problems by using simpleand similar agents that are easy to simulate in the computerwith numerical solution [4 5] Although the design ofthese kind of evolutionary algorithms can avoid complicatednumerical derivations and have good performance of globaloptimization there still exist unresolved issues Geneticalgorithm is one of the classical heuristic search approacheswhich mimics the process of natural evolution by parallelcomputing The excellent exploration ability of GA makesitself fit for intricate optimization problems but GArsquos inborndisadvantages such as slow convergence caused by mutationoperator limited the promotion of this algorithm in actualapplication Particle swarm optimization (PSO) developed byEberhart and Kennedy has received more andmore attention

regarding its potential as a global optimization techniquewith great exploitation [6ndash8] However it might be caughtin the trap of local optimization because of the prematureconvergence [5 9]

To improve further performance of these algorithmsmany variants of the original particle swarm optimizationand original genetic algorithm have been proposed [10ndash12]Furthermore a feasible method is to combine them and forma hybrid algorithm [13 14] Mutually reinforcing will makenewhybrid algorithmsmore adaptive in various optimizationproblems In the published literature different methods havebeen presented for the hybrid design of GA and PSO [2 15] Abrief summary of the relevant hybrid optimization algorithmsis described in Table 1

Although the confirmation experiment results of theseimproved algorithms perform better than the standard onesthe development of the hybrid algorithms is still limited bythe characteristic of those source algorithms The crossoverand mutation in GA only concern the position update

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 316826 12 pageshttpdxdoiorg1011552015316826

2 Mathematical Problems in Engineering

Table 1 Summary of some hybrid evolutionary algorithms and their applications

Algorithm Year Application LabelGA + PSO 2007 2008 2011 Global optimization of mathematical functions [13ndash15]GA + PSO 2012 Location and sizing of DG on distribution systems [2]NTVE-PSO 2009 The time series prediction of a practical power system [10]STHGA + GA 2010 Find the largest number of disjoint sets of sensors [36]GGA 2011 Find the optimal placement of sensors [11]A Hybrid PSO 2011 2013 Global optimization in mathematical functions [37 38]PSO using Kmeans clustering 2011 Near-global minimum-jerk joint trajectory [3]ACO + PSO 2012 A fuzzy logic controller for an autonomous mobile robot [39]CGA 2014 Nonlinear systems of second-order boundary value problems [12]PSO + SA 2014 Global optimization problems [40]

between two iterations rather than paying more attentionto agents distribution throughout the iterative process Twoor more agents usually successively visit the same regionin different iterations and they are too close to each otherto carry out effective exploration An explanation about therepeat visit problem of the agents is presented and definedas pseudo-collision (Pc) [16] Pc is inevitable in the samplingprocess for most of swarm intelligence algorithm and it isuseless and negative during exploration phase Thus thosehybrid algorithmswith GA and PSO also suffer from the neg-ative effect of Pc To reduce Pc especially during explorationphase Gao et al [16] proposed rain forest algorithm (RFA)and verified the effectiveness in benchmark mathematicalfunctions To the authors knowledge there is still no studyon the hybrid of RFA and PSO This may be because theoriginal framework of RFA is quite different from GA andother heuristic algorithms

The aim of this paper is to substitute the model of RFAfor GA tentatively by combining the global search propertyof RFA and the rapid convergence of PSO Two kinds ofagents RFAAs and PSOAs are introduced to carry out explo-ration and exploitation respectively RFAAs perform mainlyaccording to the exploration strategy of RFA and PSOAsperform according to the strategy of PSO mainly in smallerinertial factor In order to improve population diversificationuniform distribution and adaptive range division are carriedout by RFAAs in flexible scale during the iteration A furtherimprovement has been provided to enhance the convergencerate and processing speed by combining PSO algorithmwith potential guides found by both RFAAs and PSOAsSince several contingent local minima conditions may hap-pen to PSO special agent transformation is suggested toprovide information exchanging and cooperative coevolutionbetween RFAAs and PSOAs Effectiveness of the proposedmethod will be compared with several common algorithmson the various benchmark problems Finally engineeringdesign optimization problems taken from the gait control ofa simulated snake robot will be successfully solved by theproposed RFA-PSOThemain contributions of this paper aresummarized as follows

(1) A new evolutionary algorithm is developed by usingthe global search property of RFA and the rapidconvergence of PSO

(2) Population diversification is improved by both adap-tive range division and multiagents which includeRFAAs and PSOAs

(3) A further improvement on the speed and the accuracyof convergence has been provided by combining PSOalgorithmwith potential guides found by both RFAAsand PSOAs

(4) A parameter optimization problem based on centralpattern generator model for the gait control of asnake-like robot is implemented by the proposedRFA-PSO

The paper is organized as follows Section 2 will describethe hybrid algorithm of RFA-PSO To verify the performanceof RFA-PSO Section 3 presents the results obtained withthe optimization experiments on benchmark problemsThenwe apply the proposed algorithm to gait optimization of asnake-like robot and find better result in Section 4 Finallya discussion of the results is presented in Section 5

2 The Proposed Algorithm

In this section the proposed hybrid algorithm with novelRFA-PSO cooperative evolution is explained It is a combina-tion of global search property of RFA and rapid convergenceof PSO In the RFA-PSO RFAAs and PSOAs are suggestedand serve as exploration agents and exploitation agentsrespectively Special agent transformation is also suggested toprovide information exchanging and cooperative coevolutionbetween RFAAs and PSOAs

21 The Proposed Exploration Agents As exploration agentsRFAAs performmainly according to the exploration strategyof RFA Rain forest algorithm is one of the most success-ful methods with lower pseudo-collision (Pc) probabilityduring exploring phase which improves the effectivenessand efficiency of exploration carried out by the agents ofRFA It is because that little pseudo-collision outside thedominant region will reduce a large number of redundantsamples that wouldmake optimization procedure slow downAt present most of the swarm intelligence algorithms sufferfrom pseudo-collision due to the lack of relation betweensamples and the unconstrained behavior of sampling To

Mathematical Problems in Engineering 3

avoid this phenomenon partition management in globalarea and classification sampling in local area were adoptedin RFA Uniform sampling especially was proved to be themost effective exploration when the character of objectivefunction is unknown Similarly RFAAs will follow the agentsof RFA and carry out uniformdistribution and adaptive rangedivision which not only reduce the probability of pseudo-collision but also improve the population diversificationduring exploration phase

As far as possible to avoid missing the dominant areawhere there is the global optimum RFAAs will be distributeduniformly across those regions where there still are nosamples Uniform distribution will minimize sampling blankand make the whole decision space explored fully by thelimited samples But the uniform sampling suggested inthis hybrid algorithm does not mean that all the agents aredistributed uniformly throughout the optimizing processEach RFAA includes two attribute parameters 119903

119894119895119896and 119899119894119895119896

where 119903 represents the exploration range of the current agent119899 represents the number of subagents used by the currentagent for the next exploration 119894 indicates the root node orthe tree to which the current agent belongs 119895 indicates thenumber of current generation and 119896 indicates the code ofthe current agent in the 119895th generation of the 119894th tree Inthe next iteration these 119899

119894119895119896subagents will be uniformly

distributed in the range of 119903119894119895119896

around the current agent (theparent agent of these subagents) and the density of these119899119894119895119896

subagents in the range of 119903119894119895119896

indicates the degree ofexploration carried out by the parent agent RFAA

119894119895119896 Thus

the uniform sampling mentioned above refers to uniformexploration around one parent agent in local area while thedegrees of exploration carried out by each parent agent aredifferent from each other To fill the sampling blank aroundone agentwith the limited number of samples in one iterationuniform distribution is the best method which will makethe biggest sampling blank in the next iteration becomesmaller effectively Uniformly sampling in an assigned rangewill also avoid most of pseudo-collision and make full useof the limited subagents to explore unknown parameterspace

To find the potential dominant area with fewer samplesit is necessary to properly focus on the exploration inpotential dominant areas according to the previous sampleinformation The fact that RFAAs put focus on dominantareas does not mean they give up the exploration in otherareas but it shows that the focus is formed by the diversifi-cation of exploration speed RFAAs will be divided properlyinto different parts by adaptive range division The rangeparameter of dominant agents will be set to a small value andvice versa Thus the density of samples near the potentialdominant area will increase faster if each parent agent hasthe same number of subagents On the other hand the rangeparameters of the subagents are adjusted based on the statusof both their parent agents and themselves These rangeparameters also should be mainly smaller than that of theirparent agents which can avoid most of the pseudo-collisionthat arises between agents of different generation If onesubagent also get better fitness as its parent agent the focusaround this agent will be enhanced

22The Proposed ExploitationAgents As exploitation agentsPSOAs perform mainly according to the strategy of PSO insmaller inertial factor PSO is one of the most successfulmethods with higher convergence speed In PSO algorithmeach particle keeps track of its own position and velocity inthe problem space The position and velocity of a particleare initially randomly generated Then at each iteration thenew positions and velocities of the particles are updated andare convergent to the global optimum under the influenceof their personal optimum To avoid premature convergencetime-varying inertia weight is adopted in many variants ofthe canonical PSO Mostly the inertia weight will changefrommaximum values to minimum values which allows theparticles to converge towards the global optimum at the endof the search As the roles of PSOAs proposed in this paper areexploitation agents the inertia weight will be set to a smallervalue

23 The Evolution of RFAAs and PSOAs In the hybrid RFA-PSO algorithm RFAAs are updated in different density andcover the problem space widely Initially the position of basicagents is generated uniformly Then the range (119903

119894119895119896) and the

number (119899119894119895119896

) of the subagents are updated at each iterationThe updating rule of RFAAs is given by

119903119894119895119896

=

1

3

times [(1 minus 120572) times 119903119894119895minus1119896

+ 120572 times 119903119894119895minus1119896

times 1198901minus2119881119894119895minus1119896

times 119867]

(1)

119899119894119895119896

= (1 minus 120572) times 119899119894119895minus1119896

+ 120572 times 119899119894119895minus1119896


times 119867 (119899119894119895119896

ge 1)

(2)

where 120572 is a study factor that belongs to (0 1) 119881119894119895minus1119896

meansthe sampling value of the parent agent in previous iterationand 119867 is defined as an information entropy adjusting theeffect of previous sampling information 120572 can be set asmanual parameter in practice 119881

119894119895119896and 119867 are calculated

according to (1) and (4)

119881119894119895119896

=

119865119894119895119896

minus 119865min

119865max minus 119865min (3)

119867 = 1 minus 119890minus119895

(4)

where 119865119894119895119896

refers to the fitness of Agent119894119895119896

and 119865min and119865max refer to the minimum fitness and the maximum fitnessof RFAAs respectively To avoid the population of RFAAsincrease exponentially survival number (119904

119894119895) of each tree is

given by

119904119894119895


+ 120572 [119867 times min (119899119894119895119896

) + (1 minus 119867) times max (119899119894119895119896

)]

(119904119894119895

ge 1)

(5)

Thus each tree holds at least one RFAA during each iterationand the population of RFAAs will increase first and thendecrease according to the need of exploration


PSOAs perform as what the particles of PSO do in thephase of quick convergence The standard PSO algorithmemploys a population of particles initialized with randomposition and velocity The particles evolve by following thecurrent optimum particles through the problem space ofthe function to be optimized The state of each particle isrepresented by its position 119883

119894and velocity 119881

119894 Then at each

iteration the new position and velocity of each particle areupdated following two dominant positions The first one iscalled ldquo119875bestrdquo as the position of the best solution that thisparticle has achieved so far The other one is called ldquo119866bestrdquoas the best position tracked by the particle swarm so far Theupdating rule of PSO is given by

119881119895+1

119894= 120596119881119895

119894+ 11988811199031(119875best minus 119883

119895

119894) + 11988821199032(119866best minus 119883

119895

119894)

119883119895+1

119894= 119883119895

119894+ 119881119895+1

119894

(6)

where 120596 1198881 and 119888

2are three key parameters of PSO and 119903

1

and 1199032are two randomweightsThe inertia weight 120596 controls

how much the particle remembers its previous velocity Theacceleration constants 119888

1and 1198882control howmuch the particle

heads toward its personal best position and the swarmrsquos bestposition at present respectively To enhance the exploitingability of PSOAs a smaller 120596 is given in RFA-PSO andtheir initial positions are restricted within some local regionsaround the dominant agents

Special agent transformation is also suggested to pro-vide information exchanging and cooperative coevolutionbetween RFAAs and PSOAs Although PSO is capable oflocating a good solution at a significantly fast rate its ability tofine-tune the optimum solution is comparativelyweakmainlydue to the lack of diversity at the end of the evolutionaryprocess So agent transformation is suggested in this paperto improve the diversity of PSOAs with dominant RFAAsAt regular iteration intervals several inferior PSOAs will bereplaced by superior RFAAs that come from several treesin different regions Thus the position of these crowdedPSOAs will be reset at other potential dominant areas Onthe contrary the most inferior RFAA will be replaced by themost superior PSOA during every iteration All the attributesof the RFAA except for the position will be inherited bya new RFAA which is transformed from the most superiorPSOA This transformation carried out in RFA-PSO makesan excellent trade-off between exploration and exploitationwhich will improve the effectiveness and efficiency of theproposed hybrid algorithm

24 The Process of RFA-PSO The steps of RFA-PSO hybridalgorithm are explained as follows

Step 1 Setting the value of manipulative parameters such as120572 120596 119888

1 and 119888

2

Step 2 Initializing RFAAs (including the range 119903119894119895119896

thenumber 119899

119894119895119896 and the survival number 119904

119894119895of their sub-

agents) as tree uniformly through the problem space andcalculating their fitness 119865

119894119895119896according to the objective fun-

ction

Step 3 Comparing the states of the current RFAAs andcalculating their sampling value 119881

119894119895minus1119896and information

entropy 119867

Step 4 Updating the range 119903119894119895119896

and the number 119899119894119895119896

ofeach RFAA according to (1) and (2) Calculating the survivalnumber 119904

119894119895of each tree according to (5)

Step 5 Distributing new RFAAs according to the scale of thesubagents around their parent agents and calculating theirfitness 119865

119894119895+1119896 Screening out the better RFAAs based on the

survival number

Step 6 Judging the condition of carrying out PSO algorithmIf PSO is needed then continue the following steps other-wise jump to Step 3

Step 7 Initializing the swarm of PSOAs around the mostdominant RFAA within the range limit of 119903

119894101584011

Step 8 Calling the process of PSO algorithm to update thepositions of PSOAs

Step 9 Judging if special agent transformation is required Ifso take the dominant RFAAs in another tree instead of theinferior PSOAs as new PSOAs in next generation

Step 10 Judging whether to end the process of PSO If soperform the following steps otherwise jump to Step 8

Step 11 Taking the most dominant PSOA instead of the mostinferior RFAA as new RFAA in next generation Judgingwhether to end the process of RFA If so put out the optimumresult and end the program otherwise jump to Step 3

3 Benchmark Problems

In applied mathematics benchmark problems are known asartificial landscapes and are useful to evaluate characteristicsof optimization algorithms such as velocity of convergenceprecision robustness and general performance The perfor-mance of the proposed RFA-PSO algorithm was investigatedon widely used benchmark problems in the published lit-erature where other methods developed for complex non-convex optimization are compared The algorithm had beenimplemented in MATLAB 710 Four benchmark problemsas shown in Figure 1 have been chosen to illustrate theeffectiveness and efficiency of RFA-PSO by comparing withGA PSO and RFA

31 Griewank Problem The Griewank function which wasoriginally proposed by Griewank (1981) is a function widelyused to test the convergence of optimization algorithm TheGriewank function of order 119899 is defined by

119891 (119883) = 1 +

1

4000

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos(119909119894

radic119894

) (7)

The function is usually evaluated on the square 119909119894

isin

[minus100 100] for all 119894 = 1 2 It has a global minimum of ldquo0rdquo


Benchmark function Griewank

0

minus1

minus2

minus3

minus4

minus5

50

500

0minus50

minus50

minus100 minus100

X2

X1

Fitn

ess

(a) Fitness landscape of Griewank

5

5

10

10

minus5 minus5minus10 minus10

Benchmark function Needle-in-a-haystack

X2

X1

4000

3000

2000

100

0

00

Fitn

ess

(b) Fitness landscape of Needle

0

minus100

100

Benchmark function Shubert

minus300

minus200

200

X2 X1

5

5

10

10

minus5minus5

minus10 minus10

00

Fitn

ess

(c) Fitness landscape of Shubert

5

0

minus20

minus5

Benchmark function Schaffer

X2

X1

0

10

10

minus10

minus10minus15

0

minus04

minus02

minus06

minus08

minus1

Fitn

ess

(d) Fitness landscape of Schaffer

Figure 1 Fitness landscapes of benchmark functions

Table 2 Comparison of the results for the minimization on Griewank function

Number Samples number min119891(119883) (GA) min119891(119883) (PSO) min119891(119883) (RFA) min119891(119883) (RFA-PSO)1 6885 26330119890 minus 002 58160119890 minus 003 22350119890 minus 003 70870119890 minus 004

2 8921 51140119890 minus 002 11520119890 minus 002 73960119890 minus 003 12600119890 minus 006




at the point 119883 = [0 0] around which there are many localminima It is difficult for optimization algorithm to find theglobal minimum of this multimodal function directly

Table 2 and Figure 2(a) compare results of RFA-PSOagainst the other three challenging methods on benchmarkfunction of Griewank The number of samples in the tablesand the figures refers to the activities of agents on the testedalgorithm which explains the evolutionary process and thesystem resources consumed by optimization program In

each row of Table 2 the samples numbers at which RFA-PSO gets close to the global optimum are listed and theresults searched by the methods at that time are also listedin the same row It is clear that RFA-PSO outperforms otheralgorithms on multimodal space It is difficult for GA toconverge on the global optimum of ldquo0rdquo during the limititeration with 6000 samples Although PSO showed excellentconvergence it suffered frompremature convergence andwassometimes trapped in local optimum RFA-PSO and RFA


0 1000 2000 3000 4000 5000 60000

005

01

015

02

025

03

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

(a) Emergence across Griewank

0 1 2 3 4 5 6

minus3600

minus3400

minus3200

minus3000

minus2800

minus2600

minus2400

minus2200

minus2000

minus1800

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

times104

(b) Emergence across Needle

0 2000 4000 6000 8000 10000 12000 14000

minus180

minus170

minus160

minus150

minus140

minus130

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

GAPSO

RFARFA-PSO

(c) Emergence across Shubert

0 2000 4000 6000 8000 100000

0005

001

0015

002

0025

003

0035

004

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

GAPSO

RFARFA-PSO

(d) Emergence across Schaffer

Figure 2 Emergence across four benchmark functions

always found the position of global optimum 119883 = [0 0] butRFA-PSO is faster than RFA because of the effect of PSOAsput forward in this paper

32 Needle-in-a-Haystack Problem Needle-in-a-haystack(ldquoNeedlerdquo in short) is a highly unstructured problem ofwhich the function is given in (4)

119891 (119883) = minus

3

[005 + (1199092

1+ 1199092

2)]

2

minus (1199092

1+ 1199092

2)

2

(8)

For this function a low fitness-distance correlation isexpected The Needle-in-a-haystack topology where largeflat regions and four deceptive local optimums surround asingle narrowminimum is depicted in Figure 1(b) Normallyoptimizationmethods will be attracted by the localminimumand invalidated to find the global minimum out

33 Schaffer Problem The 2-dimensional Schaffer functionhas a single global optimal solution 119891(0 0) = 0 surroundedby infinite local optima and the difference between globaloptima and the suboptima is very small It has circular peaksand valleys around the global minimum point that makemany methods trapped in one of the local optima areas Thefunction of Schaffer problem is given in

119891 (119883) = 05 +

sin2 (radic1199092

1+ 1199092

2) minus 05

[1 + 0001 sdot (1199092

1+ 1199092

2)]2

(9)

The numbers of samples at which RFA-PSO detected theglobal optimum region and was close to the optimum areprovided in each row of Table 5 When RFA-PSO detects theoptimum region and gets very close to the optimum othermethods are probably trapped in local optimum or slowly getclose to the global optimum Thus RFA-PSO can find better


Table 3 Comparison of the results for the minimization on Needle function

Number Samples number min119891(119883) (GA) min119891(119883) (PSO) min119891(119883) (RFA) min119891(119883) (RFA-PSO)1 6244 minus2628 minus2633 minus3596 minus3600

2 8242 minus2852 minus3600 minus3583 minus3600




Table 4 Comparison of the results for the minimization on Shubert function

Number Samples number min119891(119883) (GA) min119891(119883) (PSO) min119891(119883) (RFA) min119891(119883) (RFA-PSO)1 1306 minus1364 minus1782 minus1814 minus18672 1462 minus1689 minus1850 minus1848 minus18673 1975 minus1765 minus1825 minus1867 minus18674 1523 minus1850 minus1818 minus1862 minus18675 2243 minus1807 minus1837 minus1847 minus1867

Table 5 Comparison of the results for the minimization on Schaffer function






results than other methods with a certain number of samplesin these five tests The emergence process across Schafferfunction can explain the comparison in detail and is shownin Figure 2(d)

Table 3 explains the samples number at which RFA-PSOreaches the global optimum and the searching result of thefour algorithms at that time Obviously both RFA-PSO andRFA can avoid the attraction of trap factors and get the globaldominant area but RFA-PSO is faster than RFA PSO canshow excellent convergence speed but it may be trapped inthe local optimum Figure 2(b) illustrates this phenomenonfurther in detail

34 Shubert Problem Thetwo-dimensional Shubert functionhas 18 global and 742 local minima in the area as what isdisplayed in Figure 6(c) The function is usually evaluated onthe square 119909

119894isin [minus10 10] for all 119894 = 1 2 The global minima

are situated in nine groups of two closely situated minimaBoth the groups and the minima inside groups are regularlyspaced The function is given in

119891 (119883) =

119899

prod

119894=1

5

sum

119895=1

119895 sdot cos [(119895 + 1) 119909119894+ 119895] (10)

As shown in Table 4 RFA-PSO is the first one to reachthe optimum minus1867 RFA like RFA-PSO can also detect theglobal dominant area but the convergence speed of RFA islower than that of RFA-PSO When RFA-PSO reaches theglobal optimum with a number of samples RFA is gettingclose to the optimum with the same number of samplesPSO shows a kind of uncertainty in these five tests It has

shown if one guiding particle can come within the globaldominant region and the optimization result might be betterthan RFA with the same number of samples AlthoughPSO can detect the global optimum region sometimes theexperience of being trapped in local optimum might slowdown the process of emergence of the optimal value GAcannot find better result than those other methods in mostof the time because of its lower convergence But its higherexploration ability benefits itself across Shubert functionwhich has multiple global optimum The exploration mayfind the global optimum region in larger probability such asthe fourth test in Table 4 The emergence process of thesealgorithms can explain the comparison results in detail andis shown in Figure 2(c)

4 Parameter Optimization of a Snake Robot

41 Optimization Problem inAnimal-Like Robots For roboticlocomotion a bioinspired approach based on central patterngenerators (CPGs) has received increasing attention regard-ing its possible simplicity of the controller and the flexibil-ity of gait generation [17ndash19] But the complex dynamicalbehavior between the robot and environment makes settingparameters in the model by trial and errors a difficult task[20] Some researchers have made attempts to apply kindsof algorithm in gait optimization in CPG-based method forlocomotion control of robots [21 22]

The model of CPG represented by differential equationsin bipedal robot locomotion usually hasmany control param-eters due to the high freedom of the biped robot an opti-mization algorithm framework based on genetic algorithm


Figure 3 Simulation model of a snake robot

is constructed to search for the optimal set of the CPGparameters which decide the motor behavior for generatingtrajectories of robot legs [23] In [24] a reinforcementlearning method for biped locomotion is studied to trainCPG in order to make the biped robot walk stably Thecontrol strategy using a particle swarm optimization (PSO)is developed and verified to be efficient to optimize theparameters of the CPG which generates the output signals ina robotic fish [25] Some researchers also try a designmethodusing an iterative technique for swimming control of roboticfish based on CPG approach

As to snake robots there is some related work that hasbeen done in parameters optimization problems In [26] asimulated snake robot with wheels is designed to analyzethe effect of some design parameters such as the numberof the links on its performance of serpentine locomotionby using statistical techniques to identify critical parametersand applying the simulated annealing algorithm to obtainoptimum setting which minimize energy consumption andmaximize traveled distance For several types of networkstructure of CPG in a snake robot [27] genetic algorithm(GA) is verified to be effective to optimize CPG parametersand synaptic weights to find optimalmoving speedThe com-bination of locomotion control approach for the robots andthe optimization algorithms mentioned above make modelsof these robots more robust and improve the adaptability ofthe robots in complex environment [28 29]

42 CPG-Based Method in a Snake Robot Engineeringdesign optimization problems taken from the gait control ofa snake-like robot are chosen to illustrate the applicabilityof RFA-PSO proposed in the paper Our approach uses alocomotion controller based on the biological concept ofcentral pattern generators (CPG) together with the proposedoptimization method RFA-PSO We present experimentswith a snake robot modeled in simulation environmentand serpentine locomotion on a horizontal ground will bediscussed The optimized results are compared with somestandard algorithms The main task of experiments is to testif RFA-PSO can find a set of optimal parameters for CPG toadapt the locomotor patterns to the properties of the practicalenvironment in a speed of serpentine locomotion as fast aspossible

A simulation model of the snake robot has been createdin Adams The simulated snake robot consists of 9 cylinderlinks (see Figure 3) Adjacent links are connected using a ring

Out

put

Out

put

2

0

minus2

2

0

minus2

u

u

0 4 8 12 16

0 4 8 12 16

t (s)

t (s)

Figure 4 Behavior of a Hopf oscillator when 119903 and 120596 are changedat time 8 s

part which forms a Cardan joint Each joint has a pitch andyaw angle limitation of 50∘ All the joints are controlled underCPG-based approach To imitate the ventral scales of realsnakes and satisfy directional friction property in serpentinelocomotion caster wheels are attached to the belly of thesnake robot and modeled with asymmetric friction whilesetting the friction coefficients 03 (friction perpendicular tothe speed) and 005 (friction parallel to the speed)

There are many nonlinear oscillators used in modelingCPG formotion control of robots such asMatsuoka oscillator[30 31] VanDer Pol oscillator [32] and amplitude-controlledoscillators [33] In this paper Hopf oscillator is adopted asan artificial pattern generator to establish a CPG model forthe snake robot due to two reasons First the harmonicoutput of Hopf oscillator is suitable for the required jointcommand signals of snake robots for achieving naturalrhythms of movements Second it has clear relationship withits parameters such as amplitude variable and phase variable

A Hopf oscillator refers to the following nonlinear differ-ential equation

= 119891119867 (119906 V) = (

120596V + (1199032minus 1199062minus V2) 119906

minus120596119906 + (1199032minus 1199062minus V2) V

) (11)

where 119909 = (119906 V)119879 and the variables 119903 and 120596 respectivelyrepresent the amplitude and the frequency of the oscilla-tor When the control parameters are modified online theoscillator will rapidly adapt to any parameter change andsmoothly converges to the modified traveling wave after ashort transient period An example of how theHopf oscillatorparameter changes can be observed in Figure 4 whenthe frequency or the amplitude is changed the oscillatorsmoothly converges to the new traveling wave

The design of the CPG is inspired from the neural circuitin the content [34] Our CPG model is based on multipleHopf oscillators with nearest neighbor coupling connectedas a chain The chain system consisting of oscillators canspontaneously produce traveling waves with constant phaselags between neighboring segments along the body from the


head to the tail of the robot This wave generated from theCPG is used to achieve serpentine locomotion on groundThe total number of oscillators is 119873 where 119873 = 8 is thenumber of actuated joints of the snake robot in simulationActuated joints are numbered 1 to 119873 from head to tail

The CPG consisting of 119873 coupled Hopf oscillators thatproduce rhythms for serpentine pattern is implemented asfollows

119894= 119891119867

(119909119894) minus 119896 119908

119894119894minus1(119909119894minus 119879119894minus1

119909119894minus1

)

+119908119894119894+1

(119909119894minus 119879119894

minus1119909119894+1

)

119879119894=

119903119894+1

119903119894

(

cos (120601119894) minus sin (120601

119894)

sin (120601119894) cos (120601

119894))

(12)

for 119894 = 2 119873 minus 1 where 119873 is the number of oscillators inCPG network 119896 is constant coupling strengths The variable119909119894is the rhythmic output signal of the 119894th oscillator The

coupling between the oscillators is defined as the weight119908119894119895 where 119908

119894119894minus1represents the weight from the (119894 minus 1)th

oscillator to the 119894th oscillator The phase biases between the(119894 minus 1)th oscillator and the 119894th oscillator are set to 120601

119894in the

direction from head oscillator to rear oscillator and to minus120601119894in

the opposite direction 119879119894is a 2D rotational transformation of

angle 120601119894

In order to reflect the symmetries of the robot andreduce the number of parameters to optimize we set severalparameters of oscillators to the same values The frequencyparameters are equal for all oscillators that is 120596

119894= 120596 The

amplitude parameters on each CPG are set as 119860 The phaseof adjacent oscillator 120601

119894is equal to Δ120601 which will determine

the phase lag between neighbored modules It is here moreconvenient to introduce Δ120601 = 2120587119870

119904119873 where 119870

119904is the

number of the S-shape for the whole snake robot The weightof the adjacent oscillators is set as119908

119894119895= 1 for all connections

With these settings the CPG asymptotically stabilizes into aset of travellingwaves that depends on the control parameterssuch as frequency 120596 amplitude 119860 and phase lag Δ120601

43TheOptimal SerpentineGait Using RFA-PSO In previoussections of the paper the proposed RFA-PSO algorithm istested in the common benchmark problems and the resultsand comparison with standard algorithms show that thehybrid RFA-PSO algorithm has better performance in thetrade-off between exploration and exploitation and improvethe reliability of getting global convergence In this sectionwe will discuss how to find optimal CPG parameters ofthe simulated snake robot by using the novel algorithmAlthough some researchers have investigated the influence ofchange of key parameters to snake robot it was still difficultto discuss optimality of the parameters due to the complexdynamics of CPG network and robot body Since it has beenshown that the frequency of oscillators has almost lineareffect on the locomotion speed [35] here we discuss thetraveling wave produced by the CPG controllers that dependson two control parameters 119860 and Δ120601 which are also theparameters to be optimized in the proposed algorithm

The move distance 119889() of the snake robot in oneperiod is suggested to represent the fitness of corresponding

Figure 5 Simulated robot crawling on a horizontal experimentalsurface at119860 = 20

∘ 120601 = 45∘(119870119904= 1)120596 = 2Hz the time step between

the snapshots is 04 s

CPG parameters As RFA-PSO is designed for minimaloptimization the objective function value can be set as theopposite number of the fitness Thus the function we wantto optimize is 119891() = minus119889() where is the parametervector containing the parameters to be optimized (oscillationamplitude 119860 and phase lag Δ120601) For each evaluation theaverage speed of the simulated robot is measured after therobot starts in the initial position andmoves in a stabilizationrun time of 20 s There are several successive cycles in onefull running and multiple sets of parameters can be testedduring the running The optimal set of parameters will besaved after comparing The move distance for a given setof parameters was calculated using the measure function inthe simulator The parameters to be optimized have beenkept into a reasonable range the amplitude between 5

∘ and50∘ and 119870

119904between 02 and 2 In experiments the two

parameters have been optimized at fixed frequencies of 120596 =

2Hz on a horizontal experimental surface Three algorithmsrain forest algorithm (RFA) genetic algorithm (GA) andparticle swarm optimization (PSO) have also been tested asa comparison to the proposed optimization algorithm withthe same parameter range where the amplitude is consideredwith a step of 5∘ and the number of the S-shape for the snakerobot is designed with a step of 02

Results of all the optimization algorithms for serpentinelocomotion are plotted in Figures 6 and 7 All the speeds ofthe experiments are in meter per second For 120596 = 2Hz theRFA-PSO algorithm found optimal parameter values of 119860 =

205∘ Δ120601 = 288

∘ (the number of S-shape for the whole snakerobot is 064) and a resulting speed of 0554ms slightlyhigher than the maximal one found with systematic testswhich was 0510ms Figure 5 shows snapshots of the ser-pentine gait at 120596 = 2Hz The maximal move speed obtainedduring systematic tests is slightly lower than the result foundby RFA-PSO because of the lack of convergent exploitationin smaller dominant area The effective exploring of RFA-PSO has successfully guided efficient exploiting towards theoptimum especially to the global optimum The optimizingprocess of RFA-PSO across simulated experiment is shownin Figures 6 and 7 At the beginning RFA-PSO started from apoor location but it found amuch better local optimal resultThen RFA-PSO jumped out of the local extremum smoothlyand found the global optimum ultimately

Figures 6 and 7 respectively describe the comparisonresults of optimizing trajectory and emergent process of RFA-PSO RFA PSO and GA across the parameters space of the


5

10

15

20

25

30

35

40

45

50

Fitness distribution and optimizing trajectory of RFA-PSO

005

01

015

02

025

03

035

04

045

05

02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50

Fitness distribution and optimizing trajectory of RFA

02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50

Fitness distribution and optimizing trajectory of PSO

02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50

Fitness distribution and optimizing trajectory of GA

02 04 06 08 10 12 14 16 18 20

Ks

A


Figure 6 Comparison of optimizing trajectory

snake-like robot The fitness distribution in the parametersspace shown by colored blocks in Figure 6 can also explainthe maximal move speed obtained during a systematic testObviously all these algorithms find some better decisionvariables which are not found by the systematic samplingPSO still shows its higher exploitation speed and quickconvergence towards the optimum in the earlier stage ofoptimizing process but the global optimum is first obtainedby RFA-PSO at about 700 samples as shown in Figure 7 GAshows more widely exploration which limits its convergencespeed In addition RFA-PSO and RFA perform better trade-off between exploration and exploitation which show fasterconvergence than GA but slower convergence than PSO It isworth noting that RFA can also jump out of local extremumand evolve towards global optimum earlier at about 1500samples as shown in Figure 7Themore accurate exploitationof both RFA-PSO and RFA in later stages benefits from their

appropriate exploration in early stages As RFA-PSO canexplore some local extremum more quickly with the help ofPSO RFA-PSO performs faster convergence than RFA andfinds the global optimum firstly

5 Conclusion

This paper presented a hybrid method of swarm evolutionunder a cooperative framework with RFAAs and PSOAsTo ensure the exploration-exploitation property of agentswe combined the global search property of RFA with therapid convergence of PSO Experimental results based onbenchmark problems suggest that the RFA-PSO algorithmperformance has better performance in different cases andis capable of locating the global optimum for all the testproblems in a reliable mannerThe performance of RFA-PSO


0 200 400 600 800 1000 1200 140004

042

044

046

048

05

052

054

056

Comparison of emergent process

Number of samples

Loco

mot

ion

spee

d (m

s)

RFA-PSORFA

PSOGA

1460 1480 1500055

0552

0554

0556

Figure 7 Comparison of emergent process

was evaluated by comparing its effectiveness and efficiencywith GA PSO and RFA The RFA-PSO detects globaldominant region and obtains better solutions more quicklythan those previously used methods In addition RFA-PSOwas also applied to engineering design optimization problemstaken from the gait control of a snake-like robot and obtainedoptimal results

These simulation results demonstrate the importance ofcooperation and division of exploration and exploitation thatwere respectively carried out by RFA and PSOThe proposedalgorithm does not simply apply RFA and PSO in differentphase but synthetically combines them together by specialagent transforming which provides information exchangingand cooperative coevolution between RFAAs and PSOAsBy this approach the performance of both RFA and PSO isimproved which is also the reason why RFA-PSO can obtainhigher effectiveness and efficiency The quick emergence ofglobal optimal results also confirms the rapidity of PSOand the reliability of RFA with uniform distribution agentsin adaptive ranges Although this paper has extended theproperty of both previous RFA and PSO by introducingtransformation agents therewill be another extendedwork inseveral directions in future studies such as how to deal withhigh dimension parameters optimization and how to simplifythe model of RFA

Conflict of Interests

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

Acknowledgments

This work was supported by the State Key Laboratory ofRobotics Shenyang Institute of Automation and funded

in part by the National Natural Science Foundation ofChina (61174027) and National High Technology Researchand Development Program (2012AA04150502) The authorsgratefully acknowledge these supports

References

[1] D C Walters and G B Sheble ldquoGenetic algorithm solution ofeconomic dispatchwith value point loadingrdquo IEEE Transactionson Power Systems vol 8 no 3 pp 1325ndash1332 1993

[2] 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

[3] H-I Liny and Y-C Liu ldquoMinimum-jerk robot joint trajectoryusing particle swarm optimizationrdquo in Proceedings of the 1stInternational Conference on Robot Vision and Signal Processing(RVSP rsquo11) pp 118ndash121 November 2011

[4] G R Harik F G Lobo and D E Goldberg ldquoThe compactgenetic algorithmrdquo IEEE Transactions on Evolutionary Compu-tation vol 3 no 4 pp 287ndash297 1999

[5] F van den Bergh andA P Engelbrecht ldquoA cooperative approachto participle swam optimizationrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 225ndash239 2004

[6] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 October1995

[7] C A Coello Coello G T Pulido andM S Lechuga ldquoHandlingmultiple objectives with particle swarm optimizationrdquo IEEETransactions on Evolutionary Computation vol 8 no 3 pp256ndash279 2004

[8] J-J Kim and J-J Lee ldquoAdaptation of quadruped gaits usingsurface classification and gait optimizationrdquo in Proceedings ofthe IEEERSJ International Conference on Intelligent Robots andSystems (IROS rsquo13) pp 716ndash721 Tokyo Japan November 2013

[9] Y-L Li W Shao L You and B-Z Wang ldquoAn improved PSOalgorithm and its application to UWB antenna designrdquo IEEEAntennas and Wireless Propagation Letters vol 12 pp 1236ndash1239 2013

[10] C-M Lee and C-N Ko ldquoTime series prediction using RBFneural networks with a nonlinear time-varying evolution PSOalgorithmrdquoNeurocomputing vol 73 no 1ndash3 pp 449ndash460 2009

[11] T-H Yi H-N Li and M Gu ldquoOptimal sensor placementfor health monitoring of high-rise structure based on geneticalgorithmrdquo Mathematical Problems in Engineering vol 2011Article ID 395101 11 pages 2011

[12] O Abu-Arqub Z Abo-Hammour and S Momani ldquoApplica-tion of continuous genetic algorithm for nonlinear system ofsecond-order boundary value problemsrdquo Applied Mathematicsamp Information Sciences vol 8 no 1 pp 235ndash248 2014

[13] F Valdez and PMelin ldquoParallel evolutionary computing using acluster for mathematical function optimizationrdquo in Proceedingsof the Annual Meeting of the North American Fuzzy InformationProcessing Society (NAFIPS rsquo07) pp 598ndash603 San Diego CalifUSA June 2007

[14] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing Journal vol 8 no 2 pp 849ndash857 2008


[15] F Valdez P Melin and O Castillo ldquoAn improved evolution-ary method with fuzzy logic for combining particle swarmoptimization and genetic algorithmsrdquo Applied Soft ComputingJournal vol 11 no 2 pp 2625ndash2632 2011

[16] W-S Gao C Shao and Q Gao ldquoPseudo-collision in swarmoptimization algorithm and solution Rain forest algorithmrdquoActa Physica Sinica vol 62 no 19 Article ID 190202 2013

[17] J Yu M Tan J Chen and J Zhang ldquoA survey on CPG-inspiredcontrol models and system implementationrdquo IEEE TransactionsonNeural Networks and Learning Systems vol 25 no 3 pp 441ndash456 2014

[18] Z Wang and H Gu ldquoA bristle-based pipeline robot for Ill-constraint pipesrdquo IEEEASME Transactions on Mechatronicsvol 13 no 3 pp 383ndash392 2008

[19] Z Wang and H Gu ldquoA review of locomotion mechanisms ofurban search and rescue robotrdquo Industrial Robot vol 34 no 5pp 400ndash411 2007

[20] H Zhao Z Wang H Shang W Hu and G Qin ldquoA time-controllable Allan variance method for MEMS IMUrdquo IndustrialRobot vol 40 no 2 pp 111ndash120 2013

[21] Z Wang J He H Shang and H Gu ldquoForward kinematicsanalysis of a six-DOF Stewart platform using PCA and NMalgorithmrdquo Industrial Robot vol 36 no 5 pp 448ndash460 2009

[22] Z Wang J He and H Gu ldquoForward kinematics analysis ofa six-degree-of-freedom stewart platform based on indepen-dent component analysis and Nelder-Mead algorithmrdquo IEEETransactions on Systems Man and Cybernetics ASystems andHumans vol 41 no 3 pp 589ndash597 2011

[23] M Oliveira V Matos C P Santos and L Costa ldquoMulti-objective parameter CPG optimization for gait generationof a biped robotrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo13) pp 3130ndash3135 Karlsruhe Germany May 2013

[24] M-A Sato Y Nakamura and S Ishii ldquoReinforcement learningfor biped locomotionrdquo in Artificial Neural NetworksmdashICANN2002 vol 2415 pp 777ndash782 Springer 2002

[25] CWang G Xie LWang andMCao ldquoCPG-based locomotioncontrol of a robotic fish using linear oscillators and reducingcontrol parameters via PSOrdquo International Journal of InnovativeComputing Information andControl vol 7 no 7 pp 4237ndash42492011

[26] H Kalani A Akbarzadeh and H Bahrami ldquoApplication ofstatistical techniques in modeling and optimization of a snakerobotrdquo Robotica vol 31 no 4 pp 623ndash641 2013

[27] K Inoue S Ma and C Jin ldquoOptimization of CPG-network fordecentralized control of a snake-like robotrdquo in Proceedings ofthe IEEE International Conference on Robotics and Biomimetics(ROBIO rsquo05) pp 730ndash735 Hongkong China July 2005

[28] Z Wang M Jiang Y Hu and H Li ldquoAn incremental learningmethod based on probabilistic neural networks and adjustablefuzzy clustering for human activity recognition by using wear-able sensorsrdquo IEEE Transactions on Information Technology inBiomedicine vol 16 no 4 pp 691ndash699 2012

[29] M Jiang H Shang Z Wang H Li and Y Wang ldquoA methodto deal with installation errors of wearable accelerometers forhuman activity recognitionrdquoPhysiologicalMeasurement vol 32no 3 pp 347ndash358 2011

[30] A Kamimura H Kurokawa E Yoshida S Murata K Tomitaand S Kokaji ldquoAutomatic locomotion design and experimentsfor a modular robotic systemrdquo IEEEASME Transactions onMechatronics vol 10 no 3 pp 314ndash325 2005

[31] C Liu Q Chen and D Wang ldquoCPG-inspired workspacetrajectory generation and adaptive locomotion control forQuadruped Robotsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 3 pp 867ndash880 2011

[32] R Ding J Yu Q Yang M Tan and J Zhang ldquoCPG-based dynamics modeling and simulation for a biomimeticamphibious robotrdquo in Proceedings of the IEEE InternationalConference on Robotics and Biomimetics (ROBIO rsquo09) pp 1657ndash1662 Guilin China December 2009

[33] A Crespi and A J Ijspeert ldquoAmphibot II an amphibious snakerobot that crawls and swims using a central pattern generatorrdquoin Proceedings of the 9th International Conference on Climbingand Walking Robots pp 19ndash27 Brussels Belgium September2006

[34] K Seo and J-J E Slotine ldquoModels for global synchronizationin CPG-based locomotionrdquo in Proceedings of the IEEE Interna-tional Conference on Robotics and Automation (ICRA rsquo07) pp281ndash286 Roma Italy April 2007

[35] C Zhou and K H Low ldquoDesign and locomotion controlof a biomimetic underwater vehicle with fin propulsionrdquoIEEEASME Transactions on Mechatronics vol 17 no 1 pp 25ndash35 2012

[36] X-M Hu J Zhang Y Yu et al ldquoHybrid genetic algorithmusing a forward encoding scheme for lifetime maximization ofwireless sensor networksrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 5 pp 766ndash781 2010

[37] M A M de Oca T Stutzle M Birattari and M DorigoldquoFrankensteinrsquos PSO a composite particle swarm optimizationalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 13 no 5 pp 1120ndash1132 2009

[38] P Melin F Olivas O Castillo F Valdez J Soria andM ValdezldquoOptimal design of fuzzy classification systems using PSO withdynamic parameter adaptation through fuzzy logicrdquo ExpertSystems with Applications vol 40 no 8 pp 3196ndash3206 2013

[39] O Castillo R Martınez-Marroquın P Melin F Valdez and JSoria ldquoComparative study of bio-inspired algorithms applied tothe optimization of type-1 and type-2 fuzzy controllers for anautonomous mobile robotrdquo Information Sciences vol 192 pp19ndash38 2012

[40] Z C Yan and Y S Luo ldquoA particle swarm optimization algo-rithm based on simulated annealingrdquo in Advanced MaterialsResearch vol 989 pp 2301ndash2305 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Table 1 Summary of some hybrid evolutionary algorithms and their applications

Algorithm Year Application LabelGA + PSO 2007 2008 2011 Global optimization of mathematical functions [13ndash15]GA + PSO 2012 Location and sizing of DG on distribution systems [2]NTVE-PSO 2009 The time series prediction of a practical power system [10]STHGA + GA 2010 Find the largest number of disjoint sets of sensors [36]GGA 2011 Find the optimal placement of sensors [11]A Hybrid PSO 2011 2013 Global optimization in mathematical functions [37 38]PSO using Kmeans clustering 2011 Near-global minimum-jerk joint trajectory [3]ACO + PSO 2012 A fuzzy logic controller for an autonomous mobile robot [39]CGA 2014 Nonlinear systems of second-order boundary value problems [12]PSO + SA 2014 Global optimization problems [40]

between two iterations rather than paying more attentionto agents distribution throughout the iterative process Twoor more agents usually successively visit the same regionin different iterations and they are too close to each otherto carry out effective exploration An explanation about therepeat visit problem of the agents is presented and definedas pseudo-collision (Pc) [16] Pc is inevitable in the samplingprocess for most of swarm intelligence algorithm and it isuseless and negative during exploration phase Thus thosehybrid algorithmswith GA and PSO also suffer from the neg-ative effect of Pc To reduce Pc especially during explorationphase Gao et al [16] proposed rain forest algorithm (RFA)and verified the effectiveness in benchmark mathematicalfunctions To the authors knowledge there is still no studyon the hybrid of RFA and PSO This may be because theoriginal framework of RFA is quite different from GA andother heuristic algorithms

The aim of this paper is to substitute the model of RFAfor GA tentatively by combining the global search propertyof RFA and the rapid convergence of PSO Two kinds ofagents RFAAs and PSOAs are introduced to carry out explo-ration and exploitation respectively RFAAs perform mainlyaccording to the exploration strategy of RFA and PSOAsperform according to the strategy of PSO mainly in smallerinertial factor In order to improve population diversificationuniform distribution and adaptive range division are carriedout by RFAAs in flexible scale during the iteration A furtherimprovement has been provided to enhance the convergencerate and processing speed by combining PSO algorithmwith potential guides found by both RFAAs and PSOAsSince several contingent local minima conditions may hap-pen to PSO special agent transformation is suggested toprovide information exchanging and cooperative coevolutionbetween RFAAs and PSOAs Effectiveness of the proposedmethod will be compared with several common algorithmson the various benchmark problems Finally engineeringdesign optimization problems taken from the gait control ofa simulated snake robot will be successfully solved by theproposed RFA-PSOThemain contributions of this paper aresummarized as follows

(1) A new evolutionary algorithm is developed by usingthe global search property of RFA and the rapidconvergence of PSO

(2) Population diversification is improved by both adap-tive range division and multiagents which includeRFAAs and PSOAs

(3) A further improvement on the speed and the accuracyof convergence has been provided by combining PSOalgorithmwith potential guides found by both RFAAsand PSOAs

(4) A parameter optimization problem based on centralpattern generator model for the gait control of asnake-like robot is implemented by the proposedRFA-PSO

The paper is organized as follows Section 2 will describethe hybrid algorithm of RFA-PSO To verify the performanceof RFA-PSO Section 3 presents the results obtained withthe optimization experiments on benchmark problemsThenwe apply the proposed algorithm to gait optimization of asnake-like robot and find better result in Section 4 Finallya discussion of the results is presented in Section 5

2 The Proposed Algorithm

In this section the proposed hybrid algorithm with novelRFA-PSO cooperative evolution is explained It is a combina-tion of global search property of RFA and rapid convergenceof PSO In the RFA-PSO RFAAs and PSOAs are suggestedand serve as exploration agents and exploitation agentsrespectively Special agent transformation is also suggested toprovide information exchanging and cooperative coevolutionbetween RFAAs and PSOAs

21 The Proposed Exploration Agents As exploration agentsRFAAs performmainly according to the exploration strategyof RFA Rain forest algorithm is one of the most success-ful methods with lower pseudo-collision (Pc) probabilityduring exploring phase which improves the effectivenessand efficiency of exploration carried out by the agents ofRFA It is because that little pseudo-collision outside thedominant region will reduce a large number of redundantsamples that wouldmake optimization procedure slow downAt present most of the swarm intelligence algorithms sufferfrom pseudo-collision due to the lack of relation betweensamples and the unconstrained behavior of sampling To




119894119895119896and 119899119894119895119896







119894119895119896 Thus





119894119895119896) and the

number (119899119894119895119896


119903119894119895119896

=

1

3


+ 120572 times 119903119894119895minus1119896


times 119867]

(1)

119899119894119895119896


+ 120572 times 119899119894119895minus1119896


times 119867 (119899119894119895119896

ge 1)

(2)





119881119894119895119896

=

119865119894119895119896

minus 119865min

119865max minus 119865min (3)

119867 = 1 minus 119890minus119895

(4)

where 119865119894119895119896




given by

119904119894119895


+ 120572 [119867 times min (119899119894119895119896

) + (1 minus 119867) times max (119899119894119895119896

)]

(119904119894119895

ge 1)

(5)





119894 Then at each


119881119895+1

119894= 120596119881119895

119894+ 11988811199031(119875best minus 119883

119895

119894) + 11988821199032(119866best minus 119883

119895

119894)

119883119895+1

119894= 119883119895

119894+ 119881119895+1

119894

(6)

where 120596 1198881 and 119888


1








1 and 119888

2


thenumber 119899





ction



entropy 119867


and the number 119899119894119895119896





survival number



119894101584011








119891 (119883) = 1 +

1

4000

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos(119909119894

radic119894

) (7)


isin




0

minus1

minus2

minus3

minus4

minus5

50

500

0minus50

minus50

minus100 minus100

X2

X1

Fitn

ess


5

5

10

10



X2

X1

4000

3000

2000

100

0

00

Fitn

ess


0

minus100

100


minus300

minus200

200

X2 X1

5

5

10

10

minus5minus5

minus10 minus10

00

Fitn

ess


5

0

minus20

minus5


X2

X1

0

10

10

minus10

minus10minus15

0

minus04

minus02

minus06

minus08

minus1

Fitn

ess













0 1000 2000 3000 4000 5000 60000

005

01

015

02

025

03

Number of samples

Obj

ectiv

e fun

ctio

n va

lue


0 1 2 3 4 5 6

minus3600

minus3400

minus3200

minus3000

minus2800

minus2600

minus2400

minus2200

minus2000

minus1800

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

times104


0 2000 4000 6000 8000 10000 12000 14000

minus180

minus170

minus160

minus150

minus140

minus130

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

GAPSO

RFARFA-PSO


0 2000 4000 6000 8000 100000

0005

001

0015

002

0025

003

0035

004

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

GAPSO

RFARFA-PSO





119891 (119883) = minus

3

[005 + (1199092

1+ 1199092

2)]

2

minus (1199092

1+ 1199092

2)

2

(8)



119891 (119883) = 05 +

sin2 (radic1199092

1+ 1199092

2) minus 05

[1 + 0001 sdot (1199092

1+ 1199092

2)]2

(9)






















119891 (119883) =

119899

prod

119894=1

5

sum

119895=1

119895 sdot cos [(119895 + 1) 119909119894+ 119895] (10)












Out

put

Out

put

2

0

minus2

2

0

minus2

u

u

0 4 8 12 16

0 4 8 12 16

t (s)

t (s)





= 119891119867 (119906 V) = (

120596V + (1199032minus 1199062minus V2) 119906


) (11)






119894= 119891119867

(119909119894) minus 119896 119908


119909119894minus1

)

+119908119894119894+1

(119909119894minus 119879119894

minus1119909119894+1

)

119879119894=

119903119894+1

119903119894

(

cos (120601119894) minus sin (120601

119894)

sin (120601119894) cos (120601

119894))

(12)





119894in the



angle 120601119894


119894= 120596 The




119904119873 where 119870

119904is the










∘ and50∘ and 119870





205∘ Δ120601 = 288




5

10

15

20

25

30

35

40

45

50


005

01

015

02

025

03

035

04

045

05

02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A





5 Conclusion



0 200 400 600 800 1000 1200 140004

042

044

046

048

05

052

054

056


Number of samples

Loco

mot

ion

spee

d (m

s)

RFA-PSORFA

PSOGA

1460 1480 1500055

0552

0554

0556






Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894119895119896and 119899119894119895119896







119894119895119896 Thus





119894119895119896) and the

number (119899119894119895119896


119903119894119895119896

=

1

3


+ 120572 times 119903119894119895minus1119896


times 119867]

(1)

119899119894119895119896


+ 120572 times 119899119894119895minus1119896


times 119867 (119899119894119895119896

ge 1)

(2)





119881119894119895119896

=

119865119894119895119896

minus 119865min

119865max minus 119865min (3)

119867 = 1 minus 119890minus119895

(4)

where 119865119894119895119896




given by

119904119894119895


+ 120572 [119867 times min (119899119894119895119896

) + (1 minus 119867) times max (119899119894119895119896

)]

(119904119894119895

ge 1)

(5)





119894 Then at each


119881119895+1

119894= 120596119881119895

119894+ 11988811199031(119875best minus 119883

119895

119894) + 11988821199032(119866best minus 119883

119895

119894)

119883119895+1

119894= 119883119895

119894+ 119881119895+1

119894

(6)

where 120596 1198881 and 119888


1








1 and 119888

2


thenumber 119899





ction



entropy 119867


and the number 119899119894119895119896





survival number



119894101584011








119891 (119883) = 1 +

1

4000

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos(119909119894

radic119894

) (7)


isin




0

minus1

minus2

minus3

minus4

minus5

50

500

0minus50

minus50

minus100 minus100

X2

X1

Fitn

ess


5

5

10

10



X2

X1

4000

3000

2000

100

0

00

Fitn

ess


0

minus100

100


minus300

minus200

200

X2 X1

5

5

10

10

minus5minus5

minus10 minus10

00

Fitn

ess


5

0

minus20

minus5


X2

X1

0

10

10

minus10

minus10minus15

0

minus04

minus02

minus06

minus08

minus1

Fitn

ess













0 1000 2000 3000 4000 5000 60000

005

01

015

02

025

03

Number of samples

Obj

ectiv

e fun

ctio

n va

lue


0 1 2 3 4 5 6

minus3600

minus3400

minus3200

minus3000

minus2800

minus2600

minus2400

minus2200

minus2000

minus1800

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

times104


0 2000 4000 6000 8000 10000 12000 14000

minus180

minus170

minus160

minus150

minus140

minus130

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

GAPSO

RFARFA-PSO


0 2000 4000 6000 8000 100000

0005

001

0015

002

0025

003

0035

004

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

GAPSO

RFARFA-PSO





119891 (119883) = minus

3

[005 + (1199092

1+ 1199092

2)]

2

minus (1199092

1+ 1199092

2)

2

(8)



119891 (119883) = 05 +

sin2 (radic1199092

1+ 1199092

2) minus 05

[1 + 0001 sdot (1199092

1+ 1199092

2)]2

(9)






















119891 (119883) =

119899

prod

119894=1

5

sum

119895=1

119895 sdot cos [(119895 + 1) 119909119894+ 119895] (10)












Out

put

Out

put

2

0

minus2

2

0

minus2

u

u

0 4 8 12 16

0 4 8 12 16

t (s)

t (s)





= 119891119867 (119906 V) = (

120596V + (1199032minus 1199062minus V2) 119906


) (11)






119894= 119891119867

(119909119894) minus 119896 119908


119909119894minus1

)

+119908119894119894+1

(119909119894minus 119879119894

minus1119909119894+1

)

119879119894=

119903119894+1

119903119894

(

cos (120601119894) minus sin (120601

119894)

sin (120601119894) cos (120601

119894))

(12)





119894in the



angle 120601119894


119894= 120596 The




119904119873 where 119870

119904is the










∘ and50∘ and 119870





205∘ Δ120601 = 288




5

10

15

20

25

30

35

40

45

50


005

01

015

02

025

03

035

04

045

05

02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A





5 Conclusion



0 200 400 600 800 1000 1200 140004

042

044

046

048

05

052

054

056


Number of samples

Loco

mot

ion

spee

d (m

s)

RFA-PSORFA

PSOGA

1460 1480 1500055

0552

0554

0556






Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894 Then at each


119881119895+1

119894= 120596119881119895

119894+ 11988811199031(119875best minus 119883

119895

119894) + 11988821199032(119866best minus 119883

119895

119894)

119883119895+1

119894= 119883119895

119894+ 119881119895+1

119894

(6)

where 120596 1198881 and 119888


1








1 and 119888

2


thenumber 119899





ction



entropy 119867


and the number 119899119894119895119896





survival number



119894101584011








119891 (119883) = 1 +

1

4000

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos(119909119894

radic119894

) (7)


isin




0

minus1

minus2

minus3

minus4

minus5

50

500

0minus50

minus50

minus100 minus100

X2

X1

Fitn

ess


5

5

10

10



X2

X1

4000

3000

2000

100

0

00

Fitn

ess


0

minus100

100


minus300

minus200

200

X2 X1

5

5

10

10

minus5minus5

minus10 minus10

00

Fitn

ess


5

0

minus20

minus5


X2

X1

0

10

10

minus10

minus10minus15

0

minus04

minus02

minus06

minus08

minus1

Fitn

ess













0 1000 2000 3000 4000 5000 60000

005

01

015

02

025

03

Number of samples

Obj

ectiv

e fun

ctio

n va

lue


0 1 2 3 4 5 6

minus3600

minus3400

minus3200

minus3000

minus2800

minus2600

minus2400

minus2200

minus2000

minus1800

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

times104


0 2000 4000 6000 8000 10000 12000 14000

minus180

minus170

minus160

minus150

minus140

minus130

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

GAPSO

RFARFA-PSO


0 2000 4000 6000 8000 100000

0005

001

0015

002

0025

003

0035

004

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

GAPSO

RFARFA-PSO





119891 (119883) = minus

3

[005 + (1199092

1+ 1199092

2)]

2

minus (1199092

1+ 1199092

2)

2

(8)



119891 (119883) = 05 +

sin2 (radic1199092

1+ 1199092

2) minus 05

[1 + 0001 sdot (1199092

1+ 1199092

2)]2

(9)






















119891 (119883) =

119899

prod

119894=1

5

sum

119895=1

119895 sdot cos [(119895 + 1) 119909119894+ 119895] (10)












Out

put

Out

put

2

0

minus2

2

0

minus2

u

u

0 4 8 12 16

0 4 8 12 16

t (s)

t (s)





= 119891119867 (119906 V) = (

120596V + (1199032minus 1199062minus V2) 119906


) (11)






119894= 119891119867

(119909119894) minus 119896 119908


119909119894minus1

)

+119908119894119894+1

(119909119894minus 119879119894

minus1119909119894+1

)

119879119894=

119903119894+1

119903119894

(

cos (120601119894) minus sin (120601

119894)

sin (120601119894) cos (120601

119894))

(12)





119894in the



angle 120601119894


119894= 120596 The




119904119873 where 119870

119904is the










∘ and50∘ and 119870





205∘ Δ120601 = 288




5

10

15

20

25

30

35

40

45

50


005

01

015

02

025

03

035

04

045

05

02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A





5 Conclusion



0 200 400 600 800 1000 1200 140004

042

044

046

048

05

052

054

056


Number of samples

Loco

mot

ion

spee

d (m

s)

RFA-PSORFA

PSOGA

1460 1480 1500055

0552

0554

0556






Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0

minus1

minus2

minus3

minus4

minus5

50

500

0minus50

minus50

minus100 minus100

X2

X1

Fitn

ess


5

5

10

10



X2

X1

4000

3000

2000

100

0

00

Fitn

ess


0

minus100

100


minus300

minus200

200

X2 X1

5

5

10

10

minus5minus5

minus10 minus10

00

Fitn

ess


5

0

minus20

minus5


X2

X1

0

10

10

minus10

minus10minus15

0

minus04

minus02

minus06

minus08

minus1

Fitn

ess













0 1000 2000 3000 4000 5000 60000

005

01

015

02

025

03

Number of samples

Obj

ectiv

e fun

ctio

n va

lue


0 1 2 3 4 5 6

minus3600

minus3400

minus3200

minus3000

minus2800

minus2600

minus2400

minus2200

minus2000

minus1800

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

times104


0 2000 4000 6000 8000 10000 12000 14000

minus180

minus170

minus160

minus150

minus140

minus130

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

GAPSO

RFARFA-PSO


0 2000 4000 6000 8000 100000

0005

001

0015

002

0025

003

0035

004

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

GAPSO

RFARFA-PSO





119891 (119883) = minus

3

[005 + (1199092

1+ 1199092

2)]

2

minus (1199092

1+ 1199092

2)

2

(8)



119891 (119883) = 05 +

sin2 (radic1199092

1+ 1199092

2) minus 05

[1 + 0001 sdot (1199092

1+ 1199092

2)]2

(9)






















119891 (119883) =

119899

prod

119894=1

5

sum

119895=1

119895 sdot cos [(119895 + 1) 119909119894+ 119895] (10)












Out

put

Out

put

2

0

minus2

2

0

minus2

u

u

0 4 8 12 16

0 4 8 12 16

t (s)

t (s)





= 119891119867 (119906 V) = (

120596V + (1199032minus 1199062minus V2) 119906


) (11)






119894= 119891119867

(119909119894) minus 119896 119908


119909119894minus1

)

+119908119894119894+1

(119909119894minus 119879119894

minus1119909119894+1

)

119879119894=

119903119894+1

119903119894

(

cos (120601119894) minus sin (120601

119894)

sin (120601119894) cos (120601

119894))

(12)





119894in the



angle 120601119894


119894= 120596 The




119904119873 where 119870

119904is the










∘ and50∘ and 119870





205∘ Δ120601 = 288




5

10

15

20

25

30

35

40

45

50


005

01

015

02

025

03

035

04

045

05

02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A





5 Conclusion



0 200 400 600 800 1000 1200 140004

042

044

046

048

05

052

054

056


Number of samples

Loco

mot

ion

spee

d (m

s)

RFA-PSORFA

PSOGA

1460 1480 1500055

0552

0554

0556






Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1000 2000 3000 4000 5000 60000

005

01

015

02

025

03

Number of samples

Obj

ectiv

e fun

ctio

n va

lue


0 1 2 3 4 5 6

minus3600

minus3400

minus3200

minus3000

minus2800

minus2600

minus2400

minus2200

minus2000

minus1800

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

times104


0 2000 4000 6000 8000 10000 12000 14000

minus180

minus170

minus160

minus150

minus140

minus130

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

GAPSO

RFARFA-PSO


0 2000 4000 6000 8000 100000

0005

001

0015

002

0025

003

0035

004

Number of samples

Obj

ectiv

e fun

ctio

n va

lue

GAPSO

RFARFA-PSO





119891 (119883) = minus

3

[005 + (1199092

1+ 1199092

2)]

2

minus (1199092

1+ 1199092

2)

2

(8)



119891 (119883) = 05 +

sin2 (radic1199092

1+ 1199092

2) minus 05

[1 + 0001 sdot (1199092

1+ 1199092

2)]2

(9)






















119891 (119883) =

119899

prod

119894=1

5

sum

119895=1

119895 sdot cos [(119895 + 1) 119909119894+ 119895] (10)












Out

put

Out

put

2

0

minus2

2

0

minus2

u

u

0 4 8 12 16

0 4 8 12 16

t (s)

t (s)





= 119891119867 (119906 V) = (

120596V + (1199032minus 1199062minus V2) 119906


) (11)






119894= 119891119867

(119909119894) minus 119896 119908


119909119894minus1

)

+119908119894119894+1

(119909119894minus 119879119894

minus1119909119894+1

)

119879119894=

119903119894+1

119903119894

(

cos (120601119894) minus sin (120601

119894)

sin (120601119894) cos (120601

119894))

(12)





119894in the



angle 120601119894


119894= 120596 The




119904119873 where 119870

119904is the










∘ and50∘ and 119870





205∘ Δ120601 = 288




5

10

15

20

25

30

35

40

45

50


005

01

015

02

025

03

035

04

045

05

02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A





5 Conclusion



0 200 400 600 800 1000 1200 140004

042

044

046

048

05

052

054

056


Number of samples

Loco

mot

ion

spee

d (m

s)

RFA-PSORFA

PSOGA

1460 1480 1500055

0552

0554

0556






Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























119891 (119883) =

119899

prod

119894=1

5

sum

119895=1

119895 sdot cos [(119895 + 1) 119909119894+ 119895] (10)












Out

put

Out

put

2

0

minus2

2

0

minus2

u

u

0 4 8 12 16

0 4 8 12 16

t (s)

t (s)





= 119891119867 (119906 V) = (

120596V + (1199032minus 1199062minus V2) 119906


) (11)






119894= 119891119867

(119909119894) minus 119896 119908


119909119894minus1

)

+119908119894119894+1

(119909119894minus 119879119894

minus1119909119894+1

)

119879119894=

119903119894+1

119903119894

(

cos (120601119894) minus sin (120601

119894)

sin (120601119894) cos (120601

119894))

(12)





119894in the



angle 120601119894


119894= 120596 The




119904119873 where 119870

119904is the










∘ and50∘ and 119870





205∘ Δ120601 = 288




5

10

15

20

25

30

35

40

45

50


005

01

015

02

025

03

035

04

045

05

02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A





5 Conclusion



0 200 400 600 800 1000 1200 140004

042

044

046

048

05

052

054

056


Number of samples

Loco

mot

ion

spee

d (m

s)

RFA-PSORFA

PSOGA

1460 1480 1500055

0552

0554

0556






Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Out

put

Out

put

2

0

minus2

2

0

minus2

u

u

0 4 8 12 16

0 4 8 12 16

t (s)

t (s)





= 119891119867 (119906 V) = (

120596V + (1199032minus 1199062minus V2) 119906


) (11)






119894= 119891119867

(119909119894) minus 119896 119908


119909119894minus1

)

+119908119894119894+1

(119909119894minus 119879119894

minus1119909119894+1

)

119879119894=

119903119894+1

119903119894

(

cos (120601119894) minus sin (120601

119894)

sin (120601119894) cos (120601

119894))

(12)





119894in the



angle 120601119894


119894= 120596 The




119904119873 where 119870

119904is the










∘ and50∘ and 119870





205∘ Δ120601 = 288




5

10

15

20

25

30

35

40

45

50


005

01

015

02

025

03

035

04

045

05

02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A





5 Conclusion



0 200 400 600 800 1000 1200 140004

042

044

046

048

05

052

054

056


Number of samples

Loco

mot

ion

spee

d (m

s)

RFA-PSORFA

PSOGA

1460 1480 1500055

0552

0554

0556






Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894= 119891119867

(119909119894) minus 119896 119908


119909119894minus1

)

+119908119894119894+1

(119909119894minus 119879119894

minus1119909119894+1

)

119879119894=

119903119894+1

119903119894

(

cos (120601119894) minus sin (120601

119894)

sin (120601119894) cos (120601

119894))

(12)





119894in the



angle 120601119894


119894= 120596 The




119904119873 where 119870

119904is the










∘ and50∘ and 119870





205∘ Δ120601 = 288




5

10

15

20

25

30

35

40

45

50


005

01

015

02

025

03

035

04

045

05

02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A





5 Conclusion



0 200 400 600 800 1000 1200 140004

042

044

046

048

05

052

054

056


Number of samples

Loco

mot

ion

spee

d (m

s)

RFA-PSORFA

PSOGA

1460 1480 1500055

0552

0554

0556






Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5

10

15

20

25

30

35

40

45

50


005

01

015

02

025

03

035

04

045

05

02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A


005

01

015

02

025

03

035

04

045

05

5

10

15

20

25

30

35

40

45

50


02 04 06 08 10 12 14 16 18 20

Ks

A





5 Conclusion



0 200 400 600 800 1000 1200 140004

042

044

046

048

05

052

054

056


Number of samples

Loco

mot

ion

spee

d (m

s)

RFA-PSORFA

PSOGA

1460 1480 1500055

0552

0554

0556






Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 1000 1200 140004

042

044

046

048

05

052

054

056


Number of samples

Loco

mot

ion

spee

d (m

s)

RFA-PSORFA

PSOGA

1460 1480 1500055

0552

0554

0556






Acknowledgments



References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article an optimization algorithm with novel rfa-pso … · 2019. 7. 31. · genetic...

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