cubic spline interpolation-based robot path...

Research ArticleCubic Spline Interpolation-Based Robot Path Planning Using aChaotic Adaptive Particle Swarm Optimization Algorithm

Jianfang Lian1 Wentao Yu 1 Kui Xiao1 and Weirong Liu2

1College of Computer and Information Engineering Central South University of Forestry and TechnologyChangsha 410004 China2Country School of Computer Science and Engineering Central South University Changsha 410083 China

Correspondence should be addressed to Wentao Yu wtyu_csuft126com

Received 15 October 2019 Revised 15 December 2019 Accepted 14 January 2020 Published 20 February 2020

Academic Editor omas Hanne

Copyright copy 2020 Jianfang Lian et al is 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

is paper proposed a cubic spline interpolation-based path planning method to maintain the smoothness of moving the robotrsquospath Several path nodes were selected as control points for cubic spline interpolation A full path was formed by interpolating onthe path of the starting point control points and target point In this paper a novel chaotic adaptive particle swarm optimization(CAPSO) algorithm has been proposed to optimize the control points in cubic spline interpolation In order to improve the globalsearch ability of the algorithm the position updating equation of the particle swarm optimization (PSO) is modified by the beetleforaging strategyen the trigonometric function is adopted for the adaptive adjustment of the control parameters for CAPSO toweigh global and local search capabilities At the beginning of the algorithm particles can explore better regions in the global scopewith a larger speed step to improve the searchability of the algorithm At the later stage of the search particles do fine searcharound the extremum points to accelerate the convergence speed of the algorithm e chaotic map is also used to replace therandom parameter of the PSO to improve the diversity of particle swarm and maintain the original random characteristics Sinceall chaotic maps are different the performance of six benchmark functions was tested to choose the most suitable onee CAPSOalgorithm was tested for different number of control points and various obstacles e simulation results verified the effectivenessof the proposed algorithm compared with other algorithms And experiments proved the feasibility of the proposed model indifferent dynamic environments

1 Introduction

Human beings always have an urge to complete all the worksand jobs automatically through automated machines whichinspires the researchers to focus on the design of mobilerobots Path planning is one of the most critical skills formobile robots It has been used in different applications suchas robot rescue [1] robot service [2] and robot patrol [3]e main goal of path planning is to find the shortest andsmooth path between the starting and target points How-ever it is hard to find the shortest and smooth collision-freepath for robots because of the complex robot workingenvironment

In general the existing robot path planning methods canbe classified into two categories classical algorithms and

heuristic optimization algorithm e main classical algo-rithms include cell decomposition artificial potential fieldand sampling-based methods [4] However classic methodsare very time consuming and require ample storage memory[5] us heuristic optimization algorithms [6 7] are usedfrequently to optimize the path planning problem such asdifferential evolutionary (DE) algorithm [8] genetic algo-rithm (GA) [9] Alowast algorithm [10] artificial bee colony(ABC) algorithm [11] annealing (SA) [12] particle swarmoptimization (PSO) [13] and ant colony optimization(ACO) [14]

GA is an optimization algorithm based on natural ge-netics which includes natural selection crossover andvariation [9] Premature convergence may occur becausegenetic algorithms operate in grid maps and do not control

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 1849240 20 pageshttpsdoiorg10115520201849240

population diversity erefore in the robot path planningapplication GA is integrated with other algorithms such asartificial potential field method artificial bee populationalgorithm etc to achieve better results [15] However themain disadvantage of improved genetic algorithms in therobot path planning field is that it takes a long time

e Alowast algorithm is an effective and direct method tosearch path in a static environment However when theAlowast algorithm was used to search in the neighborhood of thecurrent node the path beyond the threshold length isabandoned due to the local path length limitation resultingin the final path being not optimal Also if there are notenough sample points to build the graph the search successrate may be low Nevertheless increasing the number ofsampling points increases costs Hence there are manyimproved Alowast algorithms to improve the optimal range ofrobot paths and search success rates [16] ese improvedAlowast algorithms are very inefficient in dynamic environmentsBecause when the environment changes such as new bar-riers the Alowast algorithm and the improved Alowast algorithmsmust research from the current node to the end of the pathHence if the environment changes and the robotrsquos currentposition is still quite a long distance from the final targetthen the Alowast algorithm and the improved Alowast algorithms willtake a lot of time to redraw its path

e ABC algorithm is based on a simulation of theforaging behavior of the bee colony [11] ABC is easy toimplement ABC also has some drawbacks like early con-vergence stagnation and slow convergence e ABCperforms very well while exploring the feasible search spacebut it shows poor performance for exploitation BesidesABC is first applied to solve the problems of function op-timization while the path planning of a mobile robot is acombinatorial optimization problem erefore there aresome difficulties in algorithm construction

SA is a stochastic global optimization algorithm simu-lated by Kirkpatrick e convergence probability of SAalgorithm is 1 [12] is characteristic can guarantee theachievement of global optimization when it is used for robotglobal path planning while it has the disadvantages of slowconvergence speed e ACO is a random search optimi-zation algorithm with characteristics of positive feedbackand parallel computing which applies to various problemssuch as traveling salesman problem [17] quadratic pro-gramming problem [18] and production scheduling prob-lem [19] However the optimal path of ant colony algorithmplanning is time-consuming

Traditional robot path planning attempts to find anoptimal path which may contain some sharp turns andpolygonal lines However this algorithm is not flexiblebecause it adds extra workload by moving the robot alongthe sharp turn by stopping rotating or restarting evenswitching between different modes erefore traditionalrobot path planning is time-consuming and labor-intensivefor specific tasks requiring smooth motion

Spline interpolation curve is a smooth curve passingthrough a series of given points Quadratic spline interpo-lation is a quadratic polynomial interpolation When thecoefficient of the highest degree term is 0 the quadratic

spline interpolation curve is a straight line [20] ereforethe quadratic interpolation is not applicable However cubicspline interpolation has the convergence property of the firstand second derivatives When the highest term coefficient ofcubic function is 0 the cubic spline interpolation curve isstill a curve and the interpolation effect should be better Inaddition compared with high-order interpolation cubicspline interpolation has the advantages of simple calculationand good stability

Cubic spline interpolation has been widely applied Forexample cubic spline interpolation was used to control thequality of air temperature [21] By interpolating the tem-perature observations for each altitude segment suspiciousvalues in the temperature observation data can be moreeffectively marked Also cubic spline interpolation was usedto simulate the trajectory of the UAV [22] and study the pathsmoothness of the manipulator In [23] the cubic splineinterpolation was applied in the temperature compensationaspect of alcohol concentration measurement results

In this paper the moving robotrsquos path can be drawn as aseries of line segments that join the path nodes e cubicspline function was used to interpolate on the path of thestarting point control points and target point us a fullpath that was formed by connecting all interpolation pointswas obtained e control pointsrsquo position determines thelength of the path the number of which determines thenumber of spline curves and the maximum turnings ofthe patherefore these control points are used to optimizethe path of mobile robots Heuristic optimization algorithmsare an alternative to optimize control points

Particle swarm optimization (PSO) is a computationaltechnology Like other evolutionary algorithms PSO ach-ieves the search for optimal spatial solutions through in-dividual collaboration and competition However comparedwith evolutionary algorithms PSO is simple and easy toimplement Moreover PSO needs to adjust relatively fewparameters which is convenient and practical PSO has beenwidely used in many applications For example the PSOalgorithm is applied to deal with mathematical problems[24] which shows excellent performance in solving someclassical function optimization problems and even somenonlinear functions Also the PSO algorithm is introducedinto the fault diagnosis to propose a novel intelligent di-agnosis method which is applied to diagnose the faults of themotor bearing [25] en PSO is also used to solve theproblem of optimal deployment of wireless sensor network(WSN) nodes and to solve many problems caused by theoptimized deployment of static nodes [26]

However in some cases particle swarm optimizationcannot find a global optimal solution due to particle de-pletion Moreover the search strategy of the particle swarmalgorithm is mainly based on a random walk so it cannotalways successfully find the optimal solution To date somevariants of PSO have been presented to improve the originalversion PSO Most of the current existing PSOs can beroughly divided into three categories parameter selectionhybrid versions and topology structure respectively [27]

e proper selection of control parameters such asinertia weight and acceleration coefficient can significantly

2 Mathematical Problems in Engineering

influence the convergence of PSO In [28] Jiao et al pro-posed a new inertia weight particle swarm optimization(IWPSO) algorithm which uses Boltzmann search (BPSO)to adaptively adjust inertia weight C1 and C2 in the velocityupdate equation e IWPSO algorithm can guide particlesto find the most promising regions in the search space In[29] Melin et al proposed a parameter self-tuning based onfuzzy logic ey proposed a method of dynamicallyadjusting the iterative coefficients of the inertia weights C1and C2 using fuzzy control where the iterative coefficientsare the learning factors However the design of such al-gorithms is very complicated which increases the workloadMoreover the optimization accuracy of these algorithms isnot improved and the global search ability of the algorithmis weak

Many studies improve the performance of particleswarm optimization algorithms by combining particleswarm optimization with other search techniques such asdifferential evolution (DE) [30] artificial potential field(APF) [31] genetic algorithm (GA) [32] and neighborhoodsearch [33] In [34] Ding et al combined a quantum-be-havior PSO with the simplex algorithm to solve the load flowproblem However these algorithms are inherently complexso the combination of these algorithms with particle swarmoptimization algorithms will be more complicated and time-consuming

Different types of topology structures have been studiedin the literature to enhance the performance of PSO In [35]Kong et al presents a simplified particle swarm optimization(SPSO) SPSO is an improved particle swarm optimizationalgorithm that combines the particle updating formula intoone and introduces the group of optimal terms e algo-rithm performance has been improved to a certain extentHowever too many different terms in the algorithm affectthe algorithm performance and the searchability of thealgorithm cannot keep up with the convergence speed afterthe updating formula is merged

In [36] Nagra et al presented a dynamic multiswarmparticle swarm optimizer (DMSPSO) Different from PSOthe swarms in DMSPSO are dynamic and DMSPSO oftenregroups with different regrouping plans and exchangesinformation between subgroups Multipopulation particleswarm optimization (MPSO) is introduced into a nichegeneration and evolution strategy [37] Firstly PSO con-taining N particles is divided into two niche subpopulationsto generate independent search space ese niche sub-populations are not isolated in space and all the particlesevolve within their subpopulations according to PSOWhenevolution ends the two subpopulations will chase differentextremes However this kind of algorithm is complex andtime-consuming

Chaos concept is an alternative strategy to solve theproblem of particle depletion which has many excellentextrinsic properties such as periodicity ergodicity andpseudorandomness ese excellent properties are impor-tant to ensure that the generated solutions by the algorithmcan be diverse enough to reach every mode in the multi-modal objective search space potentially Hence chaossearch can escape more easily from a local optimal solution

than the standard stochastic search Also chaotic systemshave been widely used in various applications and it wasalready combined with some optimization algorithms [38]is combination may generate solutions that are moreflexible and diverse than standard optimization algorithms[39] To date there are some widely used chaotic maps suchas Singer map [40] Kent map [41] Logistic map [42] andSine map [43]

In this paper a cubic spline interpolation-based pathplanning method has been proposed to maintain thesmoothness of moving the robotrsquos path Several path nodeswere selected as control points for cubic spline interpolationA full path was formed by interpolating on the path of thestarting point control points and target point e maincontribution is to use chaotic adaptive particle swarm op-timization (CAPSO) algorithm to present a novel algorithmthat is used to optimize control points in cubic spline in-terpolation e fitness function of CAPSO synthesizes twoevaluation functions that consider path length after cubicspline interpolation and obstacle risk degree separately emain improvement of the CAPSO algorithm is illustratedbelow

To strengthen the global searchability of the algorithmwe introduced the beetle foraging strategy to modify theparticle position update formula of PSO In [44] Jiang and Liproposed the beetle antennae search algorithm according tobeetle foraging strategy e beetle antennae search algo-rithm is extremely simple and efficient erefore CAPSOimproves the searching ability without affecting the con-vergence speed

Besides the parameter ω in PSO is used to balance thesearch capabilities of local and global search of particles inthe swarm [45] e parameter C1 symbolizes self-cognitionand C2 symbolizes social influence It is advisable to take ahigher value of C1 in the beginning than C2 and graduallyreversing during the search e three control parameters inthe PSO algorithm are adaptively adjusted by using thetrigonometric function C1 and ω are both adjusted betweenlinear decrement and nonlinear decrement while C2 is ad-justed between linear increase and nonlinear increase thusat the beginning of the algorithm particles can explore betterregions in the global scope with a larger speed step to im-prove the searchability of the algorithm At the later stage ofthe search particles do excellent search around the extre-mum points to accelerate the convergence speed of thealgorithm

Since the traversal of chaotic variables is not repeatedwithin a specific range chaotic maps are used instead ofrandom parameters in the PSO algorithm After chaoticprocessing the diversity of particle group traversal is op-timized and the original random characteristics of thestandard PSO algorithm are retained which is able to ef-fectively prevent the PSO from plunging into local optimaland make the particles proceed with searching in otherregions of the solution space As different chaotic maps maylead to different behaviors of the proposed algorithm wetested the performance of six benchmark functions to choosethe most suitable one We also tested the influence of thenumber of control points on path planning Experiments

Mathematical Problems in Engineering 3

have been conducted to compare the proposed algorithmswith PSO SA GA DE and some other improved PSOalgorithms in different environments Experiments alsotested the feasibility of the proposed model in differentdynamic environments

e rest of this paper is organized as follows Section 2introduces the background Section 3 introduces theproblem formulation for robot path planning Section 4introduces the chaotic adaptive particle swarm optimizationalgorithm and Section 5 introduces the application ofCAPSO algorithm in path planning Section 6 introducesexperiments and result analysis Finally Section 7 sum-marizes the whole paper

2 Background

21 Particle Swarm Optimization PSO is inspired by thesocial behavior of some biological organisms especially thegrouprsquos ability of some animal species to locate a desirableposition in the given area In the PSO algorithm eachparticle has a memory that tracks the best position of theprevious iteration the particlersquos optimal position pbest andthe particlersquos global optimal position gbest each with a ve-locity Vi and position Xi e speed update method for theith particle at t+ 1 iteration is

vi(t + 1) ωvi(t) + c1r1(t) pbest(t) minus xi(t)1113858 1113859

+ c2r2(t) gbest(t) minus xi(t)1113858 1113859(1)

where vi(t + 1) is the new velocity of the particle at time t+ 1xi(t) is the position of the current particle at time t ωrepresents the weight C1 and C2 are the learning factors andr1 and r2 are random numbers of [0 1] which increases therandomness of particle flight e position update methodfor the ith particle at t+ 1 iteration is

xi(t + 1) xi(t) + vi(t + 1) (2)

In this paper the particle coding is coordinates of severalpath nodes in the environment e update of particlepositions in two successive iterations is shown in Figure 1

22 Cubic Spline Interpolation e path of the mobile robotshould be smooth to reduce the shaft wear of real robotsrsquowheels and energy consumption We used cubic spline in-terpolation to achieve this goal And cubic spline interpo-lation is to form a smooth curve through a series of shapepoints Take (n+ 1) nodes on the interval [a b]

a x0 ltx1 lt middot middot middot lt xn b (3)

A function f(x) on [a b] becomes an interpolated cubicspline function if the following two conditions are met

In each interval [ximinus 1 xi] f(x) is a cubic polynomialfunction

fi(x) ai + bi x minus xi( 1113857 + ci x minus xi( 11138572

+ di x minus xi( 11138573 (4)

where f(x) is continuous in the interval [a b]

f x0( 1113857 y0 f xn+1( 1113857 yn+1 (5)

fminus f+ xi( 1113857 yi i 1 2 n (6)

where fprime(x) is continuous in the interval [a b]

fminusprime xi( 1113857 f+

prime xi( 1113857 i 1 2 n (7)

where fPrime(x) is continuous in the interval [a b]

fminusPrime xi( 1113857 f+

Prime xi( 1113857 i 1 2 n (8)

In this paper cubic spline interpolation is used to in-terpolate at the starting point three path nodes and thetarget point us a completely smooth path is formed byconnecting all interpolation points

3 Problem Formulation for RobotPath Planning

Given a robot and a two-dimensional workplace includingobstacles and danger sources path planning problem istypically stated as follows to find an optimal collision-freepath from starting and ending points according to someperformance merits such as the length the time thesmoothness and the energy In this paper we pay attentionto the length and the risk degree (safety) To model the pathplanning problem we model the workplace of robots firste length can be expressed as a mathematical formula

minP(t) 1113944d

i1

xi+1 minus xi( 11138572

+ yi+1 minus yi( 11138572

1113969

(9)

where (xi yi) is a path node after interpolation and there isa total of d path nodes after interpolation In this paper theinterpolation points of each spline curve are 1000 enumber of curves is (n+ 1) where n is the number of controlpoints us the number of path nodes d is (1000lowast (n+ 1))P(t) is the sum of the lengths of adjacent path nodes after

Particle swarminfluence

gbest

pbest

vi (t + 1)

xi (t + 1)

Particle individualinfluence

Current motion influence

xi (t )

vi (t )

Figure 1 e update of particle positions in two successiveiterations


interpolation at time t which represents the length of thepath at time t

For the risk degree we suppose the obstacle is a rigidbody denoted by Ok For the sake of simplicity these ob-stacles are represented by circles and the center is Ok wherek is the number of obstacles in the problem Even with anirregular obstacle we can seek its circumscribed circle Toobtain a collision-free path the safety distance dsafe betweenthe path and the obstacle should be greater than thethreshold dmin which represents the minimum distancebetween the path and the obstacle

dsafe

xi minus Okx( 1113857

2+ yi minus Ok

y1113872 11138732

minus cOk

1113970

(10)

where Okx and Ok

y represent the horizontal and verticalcoordinates of the kth obstacle respectively and cOk rep-resents the radius of the kth obstacle e path is feasibleonly when dsafelt dmin otherwise it is not feasible e riskdegree can be expressed in a mathematical formula

P(t) infeasible path dsafe lt dmin

feasible path dsafe lt dmin1113896 (11)

e objective function of the algorithm is to achieve theabove two objectivese fitness function of CAPSO is in thefollowing mathematical formula

Min P(t)

stP(t) isin C2

P(t) isin Pfree1113896

(12)

where C2 represents a set of second-order differentiablefunctions and Pfree represents a set of collision-free pathssatisfying the constraint of formula (11)

4 Chaotic Adaptive Particle SwarmOptimization Algorithm

Although the PSO algorithm proved efficient for solvingdifferent optimization problems it still has drawbacks Insome cases particle swarm optimization cannot find a globaloptimal solution due to particle depletion Besides at thebeginning of the optimization process particles can almostwalk randomly in the entire search space while particles canwalk faster randomly in the search space which may lead tothe algorithm selecting the suboptimal solution In otherwords the search strategy of the particle swarm algorithm ismainly based on a random walk so it cannot always suc-cessfully find the optimal solution CAPSO makes the fol-lowing improvements

41 Modification Based on Chaotic Map A chaotic system issimilar to a random system but it is different from therandom phenomena that arise from the random term orcoefficient of the system itself For a real random system thevalue from a given moment does not know the determiningvalue of any subsequent moment ie the system is un-predictable in the short term For a chaotic system due to itssensitivity to the dependence of the original value its short-

term behavior is completely certain but its inherent ran-domness makes it impossible to predict the exact operationin the long run erefore chaotic systems have betterdynamics and statistical properties which is crucial to en-sure that chaotic variables can traverse all states within aspecific range without repeating In other words the chaoticorbital passes every state point in the chaotic region in afinite time ese dynamic characteristics are essential toensure that the solutions generated by the particle swarmalgorithm are sufficiently diverse In this paper the one-dimensional chaotic map is used to replace the randomparameter r1 and r2 in (1) allowing for the required mixbetween exploitation and exploration Because differentchaotic maps may lead to different behaviors of the proposedalgorithm this paper uses four well-known chaotic mapsnamely Kent Sine Singer and Logistic which constitutefour different algorithms CAPSO-Kent CAPSO-SineCAPSO-Singer and CAPSO-Logistic By comparing theadvantages and disadvantages of the four algorithms for pathplanning a mapping is selected as the optimal alternative

411 Singer Map e definition of this map is as follows

xk+1 μ 786xk minus 2331x2k + 2875x

3k minus 133x

4k1113872 1113873 (13)

where micro is a parameter of 09 to 108 in this paper a is set to101

412 Sine Map e definition of this map is as follows

xk+1 a

4sin πxk( 1113857 (14)

where 0lt alt 4 in this paper a is set to 4

413 Kent Map e definition of this map is as follows

xk+1

xk

a 0lt xlt a

1 minus xk

1 minus a altxlt 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)


414 Logistic Map e definition of this map is as follows

xk+1 axk 1 minus xk( 1113857 (16)

where 31lt alt 399 in this paper a is set to 399e speed update method for the ith particle at t+ 1

iteration is

vi(t + 1) ωvi(t) + c1xk+1 pbest(t) minus xi(t)1113858 1113859

+ c2xk+1 gbest(t) minus xi(t)1113858 1113859(17)

Figure 2 shows the chaotic value histogram of the abovefour maps with random initial values over 4096 iterationse ordinate in the figure shows the number of times thechaotic value appears in 4096 cycles As can be seen from thefigure values 0 to 1 in the Kent map occur relatively evenly


over the 4096 cycles while in the Sine map and Logistic mapvalue 0 and value 1 appear more frequently over the 4096cycles In Singer map values from 075 to 098 appear morefrequently over the 4096 cycles We use chaotic maps toupdate r1 and r2 random numbers to improve the diversity ofthe particle swarm In other words we want chaos values tobe traversed rather than one value constantly appearingHence we choose Kent map as the optimal alternative Alsoexperiments verified to select Kent map is best suitable forthe algorithm

42 Beetle Search Strategy-Based Location Update In orderto improve the searchability of the algorithm we introducedthe beetle search strategy into the position update formula ofthe particle swarm algorithme beetlersquos antennae have twoessential functions one is to detect the surrounding envi-ronment For example when an obstacle is encountered theantennae can sense its size shape and stiffnesse second isto capture the taste of food When a higher concentration ofodor is detected on one side of the antennae the beetles willrotate in the same direction Otherwise they will turn to theother side According to this simple principle beetles caneffectively find food

e main advantage of the beetle search strategy is thatits design is simple and can solve optimization problems in ashort time e beetle search strategy is as follows

Step 1 generates a random vector and normalizes it

dir rands(n 1) (18)

dir dir

norm(dir) (19)

Step 2 calculates the position of the left and rightwhiskers separately

xl(t) x(t) + d0lowastdir2

xr(t) x(t) minus d0lowastdir2

d0 step

c

(20)

where d0 is the search step size of the antennae and theratio of the step size step to the search step size d0 is afixed constant that is c is a constant Step can be a fixedvalue and a variable value is article sets it to avariable step size as a mathematical formula

step eta lowast step (21)

eta can be a constant or a variable e eta variable inthis article is a mathematical formula (19)

eta step 1lowaststep 1step 0

1113888 1113889

1 1+ 10lowastkkmax111385711138571113857((((22)

kmax is the maximum number of iterations and k iscurrent iteration times Both step 0 and step 1 areconstants eta is the coefficient of step size change andless than 1 In formula (22) eta decreases with theincrease of k A large search step size means a largesearch areaus at the beginning of the algorithm thelarger search step size can explore better areas in theglobal scope and in the late stage of the search thesmaller step size can do a good search near the ex-tremum point

05

10152025303540

(a)

0

10

20

30

40

50

(b)

0

10

20

30

40

50

(c)

05

101520253035404550

(d)

Figure 2 (a) Kent chaotic value histogram (b) Sine chaotic value histogram (c) Logistic chaotic value histogram (d) Singer chaotic valuehistogram


Step 3 calculates the odor intensity of the left and rightwhiskers separately

fleft f(xl)

fright f(xr)(23)

Step 4 compares the odor intensity of the left and rightwhiskers and updates the position of the beetle

xb(t + 1) xb(t) + step lowast sign(xl(t) minus xr(t)) fl(t)lefr(t)

xb(t) minus step lowast sign(xl(t) minus xr(t)) fl(t)gtfr(t)1113896 (24)

which is

xb(t + 1) xb(t) minus step lowast dir lowast sign(fl(t) minus fr(t))

(25)

In order to improve the global search ability of thealgorithm e position of the particle in CAPSO isupdated as follows

xi(t + 1) xi(t) + M]i(t + 1) +(1 minus M)xb(t) (26)

where M is a constant in this paper M is set to 034

43 Trigonometric Function Adaptation PSO includes threeparts current motion influence individual particle influ-ence and particle swarm influence e first part is con-trolled by the weighting factor ω e second and third partsare controlled by the acceleration factors C1 and C2 A largerω is useful for jumping out of the local optimal while asmaller ω is suitable for the algorithm to converge eoptimal value of the particle swarm is crucial in the earlystage of algorithm optimization while the optimal valueof particle individual is significant in the later stageerefore in this paper the inertia weight factor and theacceleration factor are adaptively adjusted by using thetrigonometric function at each stage of the algorithmoperation

ω(k) ωmax minus ωmin

2lowast cos

πlowast k

kmax1113888 1113889 +

ωmax + ωmin( 1113857

2 (27)

where kmax is the number of final iterations k is the numberof iterations of the algorithm and ω(k) is the inertiaweighting factor corresponding to the kth iteration In thispaper kmax is set to 100 ωmax is set to 09 and ωmin is set to04 e changes in the weighting factor in the algorithmoperation are shown in Figure 3 where the maximumnumber of iterations is 500

C1(k) cosπlowast k

kmax1113888 1113889 + a (28)

C2(k) minus cosπlowast k

kmax1113888 1113889 + a (29)

where a is set to 15 e changes in the three parameters inthe algorithm operation are shown in Figure 1 where themaximum number of iterations is 500

e parameter ω in PSO is used to balance the searchcapabilities of local and global search of particles in theswarm e parameter C1 symbolizes self-cognition and C2symbolizes social influence It is advisable to take a highervalue of C1 in the beginning than C2 and gradually reversingduring the search In other words a higher value of C1 ahigher value of ω and a lesser value of C1 are advisable at thebeginning of the algorithm while a lesser value of C1 a lesservalue of ω and a higher value of C2 are advisable at the laterstage of the search e three control parameters in the PSOalgorithm are adaptively adjusted by using the trigonometricfunction C1 and ω are both adjusted between linear dec-rement and nonlinear decrement while C2 is adjusted be-tween linear increase and nonlinear increase

As shown in Figure 3 at the beginning of the algorithmAgtB Egt FgtGgtKgt L Also CgtD MgtNgtOgt PgtQ atthe later stage of the search us changing to the trigo-nometric functions make it better compared to the linearadaptation and constant At the beginning of the algorithmparticles can explore better regions in the global scope with amore massive speed step to improve the searchability of thealgorithm At the later stage of the search particles doexcellent search around the extremum points to acceleratethe convergence speed of the algorithm

5 The Application of CAPSO Algorithm inPath Planning

rough the introduction of the above sections this sectionproposes the CAPSO algorithm and wants to compare thetraditional PSO algorithm and the CAPSO algorithm has ahigher global searchability and search speed Figure 4 showsthe flowchart of CAPSO in robot path planning

e path planning process based on the CAPSO algo-rithm is as follows

Step 1 initialize the particle group including thepopulation size N and the velocity Xi position of eachparticle ViStep 2 calculate the fitness value of each particle fit[i]Step 3 for each particle compare its fitness value fit[i]with the individual extremum pbest[i] If fit[i]lt pbest[i]replace pbest[i] with fit[i]Step 4 for each particle compare its fitness value fit[i]with the global extremum gbest If fit[i]ltgbest[i] replacegbest[i] with fit[i]


Start

Initialize the parameters of CAPSO particle velocity and position

Calculate the individual optimum

Calculate the population optimum

Update r1 and r2

Update the velocity of the particle

Update the particlersquos position

Satisfy the end condition

Produce path

Output result

End

Boundary processing

Calculate the fitness value of the path

Update the path

Y

N

Updated parameter

Logistic map

Sine map

Singer map

Kent map

Choose a map

Figure 4 Flowchart of robot path planning based on CAPSO algorithm

100 200 300 400 5000Number of iterations

04

05

06

07

08

09 W

val

ue

A

B

C

D

Trig function adaptation Linear adaptation

(a)


05

1

15

2

25

Val

ue

E

F

G

K

L

M

N

O

P

Q

Trig function adaptive C2Linear adaptive C2Constant of C1 or C2

Trig function adaptive C1Linear adaptive C1

(b)

Figure 3 (a) e variation diagram of ω in algorithm operation (b) e variation diagram of C1 and C2 in algorithm operation


Step 5 iteratively update the position of the particleaccording to formula (26)Step 6 use the chaotic map to update r1Step 7 iteratively update the velocity of the particleaccording to updated formula (17)Step 8 boundary condition processingStep 9 update parameters according to formulae(27)ndash(29)Step 10 determine whether the algorithm terminationcondition is satisfied if yes end and output the opti-mization result otherwise return to Step 2

6 Experiments

In this section the performance of the proposed CAPSOalgorithm is evaluated by experiments and the optimal pathplanning method based on CAPSO is obtained In the firstsection the experimental environment and parameter settingsare given In the second section the performances of CAPSOalgorithms with different chaotic maps were tested In thesecond section the performance of the CAPSO algorithm forselecting chaotic optimal substitution and traditional heuristicalgorithms was tested In the third section the performance ofthe CAPSO algorithm for selecting chaotic optimal replace-ment and improved PSO algorithms was tested

61 Experimental Environment and Parameter Settingse environment or workplace was two-dimensional Inorder to get an unbiased comparison of CPU time all ex-periments were performed on the same PC In the envi-ronment 1 the domains of x and y were between 0 and 15Meanwhile the positions of the start point and the endpointwere (15 65) and (72 12) respectively so the straight-linedistance between the start point and the endpoint is 77833ere are five circular obstacles scattered in the environ-ment ey are located on [xobs yobs] wherexobs = 21 36 50 601113858 1113859 and yobs = 52 25 45 201113858 1113859 eradius of the obstacles is set to 07 07 10 071113858 1113859

In environment 2 the domains of x and ywere between minus 5and 30 Meanwhile the positions of the start point and theendpoint were (0 0) and (20 20) respectively so the straight-line distance between the start point and the endpointis 282843 ere are nine rectangular obstacles scattered inthe environment eir outer circle is located on [xobs yobs]where xobs 3 4 45 7 10 115 155 171113858 1113859 and yobs

3 17 9 5 12 5 9 161113858 1113859 e radius of the obstacles robs isset to 14142 35356 15811 14141 44721 11180 11180 223601113858 1113859

In environment 3 we set up 100 random collision-freeobstacles e domains of x and y were between minus 200 and1300 Meanwhile the positions of the start point and theendpoint were (0 0) and (1000 1000) respectively so thestraight-line distance between the start point and the end-point is 141421365

In environment 4 the domains of x and ywere between minus 4and 24 Meanwhile the positions of the start point and theendpoint were (0 0) and (12 13) respectively so the straight-line distance between the start point and the endpoint is

176918 ere are nine rectangular obstacles scattered in theenvironment eir outer circle is located on [xobs yobs]where xobs 2 2 5 7 7 9 95 1051113858 1113859 and yobs

2 8 6 275 105 8 1 1051113858 1113859 e radius of the obstaclesrobs is set to 14142 35356 15811 14141 44721 11180 11180 22360[ ]

In environment 5 first we set up 50 random collision-free obstacles and then randomly selected random obstaclesto make them disappear e domains of x and y werebetween 50 and 1100 Meanwhile the positions of the startpoint and the endpoint were (0 0) and (1000 1000) re-spectively so the straight-line distance between the startpoint and the endpoint is 141421365

In environment 6 first we set up four static obstaclesand four dynamic obstacles which move in a straight linerespectively e domains of x and y were between 0 and 12Meanwhile the positions of the start point and the endpointwere (0 0) and (12 10) respectively so the straight-linedistance between the start point and the endpoint is 156205

In environment 7 we set up 30 random collision-freeobstacles and then randomly selected random obstacles tomake them disappear Besides the destination vertex israndomly transformed three times during the algorithm run

e parameters of the CAPSO algorithm are as followsthe number of particles is Pop which is set to 30 emaximum velocity of the particles is VelMax which is set to512 e minimum velocity VelMin is set to minus 512 c is set to20 and the step is set to 200 Step 0 and Step 1 are set to 029and 024 respectively

62 First Experiment Comparison between Different CAPSOAlgorithms In this section three groups of test functions withdifferent characteristics are used to benchmark the perfor-mance of the proposed different CAPSO algorithms which areunimodal functions [46 47] multimodal functions [48 49]and fixed-dimension multimodal functions [50 51] especific form of the function is given in Table 1 where Dimrepresents the dimension of the function Range represents therange of independent variables that is the range of populationand fmin represents the minimum value of the function

Figure 5 shows the two-dimensional versions of a uni-modal function multimodal function and fixed-dimensionmultimodal function respectively e unimodal testfunction has only one global optimal solution which ishelpful to find the global optimal solution in the searchspace and it can test the convergence speed and efficiency ofthe algorithm thoroughly while the multimodal functionand the fixed-dimension multimodal test function havemultiple local optimal solutions which can be used to testthe algorithm to avoid the performance of the optimal localsolution and the fixed-dimension multimodal functioncompared with unimodal test function is more challengingConvergence curves of CAPSO-Kent CAPSO-SineCAPSO-Singer and CAPSO-Logistic are compared inFigure 6 for all of the test functions e figure shows thatCAPSO-Kent has good processing ability for unimodalfunctions multimodal functions and fixed-dimensionfunctions and the processing process is very stable Espe-cially when solving more complex fixed-dimension


functions CAPSO-Kent shows a more obvious advantagethan other algorithmserefore we choose the Kent map torepresent r1 and r2

63 Second Experiment Different Numbers of Control Pointsis experiment aims to test the influence of the number ofcontrol points on the proposed algorithm CAPSO with Kentmap In this experiment the number of control pointsranged from one to ten control points e population sizewas 30 Figure 7 shows the results of experiment 2 AlsoFigure 8 shows the convergence curve of the CAPSO al-gorithmwhen the number of control points ranged from oneto nine and when the population size was 30

As shown the best results were achieved when the numberof control points was small ie n 3 or 4 and the resultsdramatically decreased when the value of nwas more than fiveand the worst results were achieved when n 8 and n 9 elarger the number of control points the larger the search spaceHence the larger population is required to obtain good resultsAlso when the number of control points is small too few turnsof the curve will limit the choice of paths such as n 1 or n 2

erefore three to six control points are relatively matureWhen the number of obstacles increases sharply the number ofcontrol points can be increased appropriately

64 =ird Experiment CAPSO versus Traditional HeuristicAlgorithms in Path Planning is section tested the pathplanning of robots in environment 2 based on GA [9] SA[12] PSO [13] and CAPSO algorithms For a fair com-parison the population size in all algorithms the number ofcontrol points and the maximum number of iterations areset to 30 3 and 500 respectively Each experiment ran 15000objective function evaluations equivalent to 500 iterationsusing 30 population sizes For each algorithm a total of 25runs were performed for each experiment Figure 9(a) showsthe best paths in environment 2 generated by the CAPSOPSO GA DE and SA during 25 trialse yellow square andgreen pentacle in Figure 9(a) are the start point and endpointof the path respectively Also Figure 9(b) shows the con-vergence curves of the best fitness values during 25 trials

Based on the three factors of solution quality stability andconvergence speed the performances of these algorithms

Table 1 Description of benchmark functions

Characteristic Function Dim Range fmin

Unimodal functionsF1(x) 1113936

ni1 x2

i 30 [minus 30 30] 0F2(x) 1113936

ni1 |xi| + 1113937

ni1 |xi| 30 [minus 30 30] 0

F3(xi) 1113936ni1[x2

i minus 10 cos(2πxi) + 10] 30 [minus 512 512] 0Multimodal functions F4(x) minus 20 exp(minus 02

(1n) 1113936

nij x2

i

1113969) minus exp((1n) 1113936

ni1 cos(2πi) + 20 + e 30 [minus 32 32] 0

Fixed-dimension multimodalfunctions

F5(x) 4x21 minus 21x4

1 + (13)x61 + x1x2 minus 4x2

2 + 4x41 2 [minus 5 5] 0

F6(x) minus 11139364i1 Ci exp(minus 1113936

3j1 aij(xj minus pij)) 3 [minus 1 3]

005

1

15

2

10050

0ndash50

ndash100

100500ndash50ndash100

0020406081121416182

x2 x1

times104

times104

(a)

020406080

100

6 4 2 0 ndash2 ndash4 ndash6

6420ndash2ndash4ndash6

10

20

30

40

50

60

70

80

x2 x1

(b)

060

100

40

200

8020 60

300

400

400

200

500

ndash20 ndash20ndash40 ndash40ndash60ndash60 ndash80

50100150200250300350400450

x2 x1

(c)

Figure 5 (a) 2D version of unimodal function (b) 2D version of multimodal function (c) 2D version of fixed-dimension multimodal function


0 100 200 300 400 500Number of iterations

CAPSO-KentCAPSO-Sine

CAPSO-LogisticCAPSO-Singer

103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(a)




103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(b)




25

20

15

10

5

0

Fitn

ess v

alue

(c)




101

100

Fitn

ess v

alue

(d)




102

101

100

Fitn

ess v

alue

(e)




7

6

5

4

3

2

1

0

Fitn

ess v

alue

(f )

Figure 6 (andashf) Comparison of convergence curves of different CAPSO algorithms obtained in the benchmark function (F1ndashF6)


were compared e solution quality can be expressed by theaverage optimal fitness value which is defined as the average of25 optimal fitness values produced by 25 trials e smaller theaverage optimal fitness value is the higher the solution qualityis e stability of the algorithm is determined by the standarddeviatione smaller the standard deviation is themore stablethe algorithm is e convergence speed of the algorithm isdetermined by the number of iterations required for the al-gorithm to converge to the optimal or suboptimal solutionGiven the maximum number of iterations the fewer iterationsthe algorithm converge to the optimal solution the faster theconvergence speed

As we can see although path planners based on fivealgorithms can successfully generate a collision-avoidancepath CAPSO found the near-optimal path and performsbetter e path planners based on DE and SA have slow

convergence speed In contrast the path planners based onPSO CAPSO and GA have fast convergence speedMoreover the solution quality of path planners based on DEand SA is inferior to that based on PSO CAPSO and GA

In 25 trials the complete performance analysis for the bestperformance of different algorithms in path planning ispresented in Table 2 From the statistical data in Table 2 it canbe seen that the CAPSO-based path planner has the smallestaverage fitness value execution time and standard deviationwhich further proves the high searchability simplicity androbustness of CAPSO In addition the number of optimaliterations of CAPSO is the smallest which further proves therapid convergence of CAPSO Compared with other algo-rithms the CAPSO algorithm has the shortest execution timeFrom the six statistics listed in Tables 3 and 4 we proved thehigh searchability simplicity and robustness of CAPSO Also

656

555

454

353

252

151 2 3 4

n = 15 6 7

(a)

656

555

454

353

252

151 2 3 4 5 6 7

n = 2

(b)

656

555

454

353

252

151 2 3 4 5 6 7

n = 3

(c)

656

555

454

353

252

151 2 3 4 5 6 7

n = 4

(d)

656

555

454

353

252

151 2 3 4 5 6 7

n = 5

(e)

656

555

454

353

252

151 2 3 4 5 6 7

n = 6

(f )

656

555

454

353

252

15

1 2 3 4 5 6 7n = 7

(g)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 8

(h)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 9

(i)

Figure 7 Path planning results of CAPSO under different numbers (n 1sim9) of control points in environment 1


CAPSO is not time-consuming All in all the performance ofCAPSO is better than the other five tested algorithms

65 Fourth Experiment CAPSO versus Improved PSO Algo-rithms in Path Planning e experiment was divided into

two parts according to two types of environments (envi-ronment 3 and environment 4) is section tested the pathplanning of robots based on FAPSO [29] IWPSO [28]DEPSO [30] SPSO [35] MPSO [37] and CAPSOalgorithms

80

70

60

50

40

30

20

10

00 20 40 60 80 100

Number of iterations

Fitn

ess v

alue

n = 1n = 2n = 3n = 4n = 5

n = 6n = 7n = 8n = 9

Figure 8 Convergence curves of CAPSO with different numbers of control points in environment 1

0 5 10 15 200

5

10

15

20

PSOCAPSOGA

DESA

(a)


25

30

35

40

45

50

55Fi

tnes

s val

ue

PSOCAPSOGA

DESA

(b)

Figure 9 (a) Comparison of path planning results in environment 2 (b) Convergence curves of different algorithms in environment 2

Table 2 Path planning results in environment 2

Algorithms Best fitnessvalue

Mean fitnessvalue

Worst fitnessvalue

Average executiontime

Standarddeviation

Number of optimaliterations

DE 328655 338655 350157 123581 11001 329GA 314532 324532 336302 56875 10895 50SA 322260 359860 341927 261047 09834 396PSO 300592 320592 318083 50907 08744 50CAPSO 297213 299821 311723 47042 07255 44


(1) Example 1In this example the population size in allalgorithms the number of control points and themaximum number of iterations are set to 30 7 and100 respectively Each experiment ran 3000 ob-jective function evaluations equivalent to 100 it-erations using 30 population sizes For eachalgorithm a total of 25 runs were performed foreach experiment Figure 10(a) shows the best pathsin environment 3 generated by the CAPSO (greenline) DEPSO (black line) MPSO (red line) SPSO(rose red line) FAPSO (blue line) and IWPSO(yellow line) during 25 trials e yellow square andgreen pentacle in Figure 10(a) are the start pointand endpoint of the path respectively AlsoFigure 10(b) shows the convergence curves of thebest fitness values during 25 trials

(2) Example 2 In this example the population size in allalgorithms the number of control points and themaximum number of iterations are set to 30 4 and200 respectively Each experiment ran 6000 objectivefunction evaluations equivalent to 200 iterationsusing 30 population sizes For each algorithm a totalof 25 runs were performed for each experimentFigure 11(a) shows the best paths in environment 4generated by the CAPSO (red-brown line) DEPSO(orange line) MPSO (purple line) SPSO (blue line)FAPSO (green line) and IWPSO (black line) during25 trials e yellow square and green pentacle inFigure 11(a) are the start point and endpoint of thepath respectively And Figure 11(b) shows the con-vergence curves of best fitness values during 25 trials

As shown in Figures 10 and 11 although path planners basedon six algorithms can successfully generate a collision-avoidancepath CAPSO found the near-optimal path and performs betterAlso all paths are smooth and there are no large corners

In 25 trials the mean fitness value and standard deviationin environment 3 and environment 4 are shown in Tables 3

and 4 From the statistical data in Tables 3 and 4 we can seethat the CAPSO-based path planner has the smallest averagefitness value execution time and standard deviation whichfurther proves the high searchability simplicity and ro-bustness of CAPSO In addition the number of optimal it-erations of CAPSO is the smallest which further proves therapid convergence of CAPSO Also CAPSO is the algorithmwith the shortest time compared with other algorithms Fromthe six statistics listed in Tables 3 and 4 we proved the highsearchability simplicity and robustness of CAPSO AlsoCAPSO is not time-consuming All in all the performance ofCAPSO is better than the other five tested algorithms

66 Fifth Experiment CAPSO Algorithms in Dynamic PathPlanning under Environment 5 is section tested thefeasibility of the proposed model in the dynamic environ-ment 5 Figures 12(a)ndash12(d) show the best paths in envi-ronment 5 generated by the CAPSO Figure 13 shows theconvergence curves of the best fitness values

As we can see the overall path is globally optimal at thesame time the local path is smoother at each stage thecorner is small and gentle and it conforms to the mobilecontrol of the robot and there is no obstacle unobstructednear the target point

67 Sixth Experiment CAPSO Algorithms in Dynamic PathPlanning under Environment 6 is section tested thefeasibility of the proposed model in dynamic environment 6which contains four static obstacles and four dynamic ob-stacles Static obstacles are represented by filling graphswhile dynamic obstacles are represented by hollow bluecircles Figures 14(a)ndash14(f ) show the best paths in envi-ronment 6 generated by CAPSO

Aswe can see the overall path is globally optimal at the sametime the local path is smoother at each stage the corner is smalland gentle and it conforms to the mobile control of the robotand there is no obstacle unobstructed near the target point



Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


CAPSO 141443 142983 141624 151812 09064 46DEPSO 169577 173187 169776 410869 09968 73FAPSO 172620 182328 172899 190295 13953 35SPSO 172287 180255 172590 169308 15131 45IWPSO 181476 188891 181682 266719 10275 74MPSO 161044 171044 161383 194110 17239 80



Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


CAPSO 184155 197854 200875 71129 08360 46DEPSO 190657 208703 208497 159505 0 8920 39FAPSO 205181 224203 225167 139275 09993 35SPSO 205665 228267 227347 94258 10841 41IWPSO 203603 215655 223353 91382 09875 42MPSO 188765 205745 212343 772038 11789 21


0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue

10 20 30 40 50 60 70 80 90 1000Number of iterations

MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion

0 20 40 60 80 100 120 140 160 180 200Number of iterations

CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)



68 Sixth Experiment CAPSO Algorithms in DynamicPath Planning under Environment 7 is section testedthe feasibility of the proposed model in dynamic envi-ronment 7 which contains four dynamic obstacles

and dynamic object vertex Obstacles are representedby hollow blue and red circles Figures 15(a)ndash15(d)show the best paths in environment 7 generated byCAPSO

1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)

Figure 12 (andashd) Dynamic Path planning results in environment 5


1000

2000

3000

4000

5000

6000

Fitn

ess v

alue

Figure 13 Convergence curves of CAPSO algorithms in environment 5


12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )

Figure 14 (andashf) Dynamic path planning results in environment 6

500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions

is paper used the chaotic adaptive particle swarm optimi-zation (CAPSO) algorithm to present a novel algorithmwhich isused to optimize Cubic spline interpolation to get the shortestsmooth and collision-free path e main improvement of thisalgorithm is illustrated belowe CAPSO is an improved PSOalgorithm It introduces the beetle foraging strategy to modifythe particle position updating equation to strengthen the globalsearch ability of the algorithm en the three control pa-rameters in the PSO algorithm are adaptively adjusted by usingthe trigonometric function to make CAPSO adaptively adaptcontrol parameters between linearly decreasing and nonlinearlydecreasing strategies so that control parameters of CAPSOreach the optimal level in the iterative process Also the chaoticmap is used to replace the random parameter of the PSO toimprove the diversity of particle swarm and maintain theoriginal random characteristics Experiments show that theKent map is the most suitable With CAPSO the global searchability and search speed are improved and optimal robot pathplanning under static environment is realized Moreover ex-periments verified the feasibility of the proposed model indifferent dynamic environments e simulation results provedthe effectiveness and robustness of the proposed algorithm

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that the research was conducted in theabsence of any commercial or financial relationships thatcould be construed as a potential conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation (grant nos 61602529 and 61672539) PlannedScience and Technology Project of Hunan Province (grantno 603286580059) and Scientific Innovation Fund forGraduates of Central South University of Forestry andTechnology (grant no CX20192077)

References

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[2] X Liu L Wang D Wang et al ldquoResearch on substation PDmonitoring method based on patrol robotrdquo in Proceedings ofthe 2019 International Conference on High Performance BigData and Intelligent Systems (HPBDampIS) Shenzhen ChinaMay 2019

[3] C Feng X Sun and D Wei ldquoInertia weight particle swarmoptimization with Boltzmann explorationrdquo in Proceedings of the2011 Seventh International Conference on Computational Intel-ligence and Security Sanya Hainan China December 2011

[4] T M Cabreira C D Franco P R Ferreira andG C Buttazzo ldquoEnergy-aware spiral coverage path planningfor uav photogrammetric applicationsrdquo IEEE Robotics andAutomation Letters vol 3 no 4 pp 3662ndash3668 2018

[5] J Lian W Yu andW Liu ldquoA chaotic adaptive particle swarmoptimization for robot path planningrdquo in Proceedings of the2019 Chinese Control Conference (CCC) Guangzhou ChinaJuly 2019

[6] L K Panwar S Reddy K A Verma B K Panigrahi andR Kumar ldquoBinary grey wolf optimizer for large scale unitcommitment problemrdquo Swarm and Evolutionary Computa-tion vol 38 pp 251ndash266 2018

[7] PanwarL Kumar et al ldquoOptimal schedule of plug in electricvehicles in smart grid with constrained parking lotsrdquo inProceedings of the 6th International Conference on PowerSystems (ICPS) New Delhi India March 2016

[8] M Wang J Luo J Fang and J Yuan ldquoOptimal trajectoryplanning of free-floating space manipulator using differentialevolution algorithmrdquo Advances in Space Research vol 61no 6 pp 1525ndash1536 2018

500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)

Figure 15 (andashd) Dynamic path planning results in environment 7


[9] R Roy M Mahadevappa and C S Kumar ldquoTrajectory pathplanning of eeg controlled robotic arm using GArdquo ProcediaComputer Science vol 84 pp 147ndash151 2016

[10] F Duchon A Babinec M Kajan et al ldquoPath planning withmodified a star algorithm for a mobile robotrdquo Procedia En-gineering vol 96 pp 59ndash69 2014

[11] H Liu B Xu D Lu and G Zhang ldquoA path planning ap-proach for crowd evacuation in buildings based on improvedartificial bee colony algorithmrdquo Applied Soft Computingvol 68 pp 360ndash376 2018

[12] R S Tavares T C Martins and M S G Tsuzuki ldquoSimulatedannealing with adaptive neighborhood a case study in off-linerobot path planningrdquo Expert Systems with Applicationsvol 38 no 4 pp 2951ndash2965 2011

[13] H-T Hsieh and C-H Chu ldquoImproving optimization of toolpath planning in 5-axis flank milling using advanced PSOalgorithmsrdquo Robotics and Computer-Integrated Manufactur-ing vol 29 no 3 pp 3ndash11 2013

[14] S Gulcu M Mahi O K Baykan and H Kodaz ldquoA parallelcooperative hybrid method based on ant colony optimizationand 3-Opt algorithm for solving traveling salesman problemrdquoSoft Computing vol 22 no 5 pp 1669ndash1685 2018

[15] M Elhoseny A arwat A E Hassanien and Aboul EllaHassanien ldquoBezier curve based path planning in a dynamicfield using modified genetic algorithmrdquo Journal of Compu-tational Science vol 25 pp 339ndash350 2018

[16] J Yao ldquoPath planning for virtual human motion using im-proved Alowast star algorithmrdquo in Proceedings of the SeventhInternational Conference on Information Technology NewGenerations Washington DC USA April 2010

[17] M Mahi O K Baykan and H Kodaz ldquoA new hybrid methodbased on particle swarm optimization ant colony optimiza-tion and 3-opt algorithms for traveling salesman problemrdquoApplied Soft Computing vol 30 pp 484ndash490 2015

[18] M Zangari A Pozo R Santana and A Mendiburu ldquoAdecomposition-based binary ACO algorithm for the multi-objective UBQPrdquo Neurocomputing vol 246 pp 58ndash68 2017

[19] S Yu J Zhang S Zheng and H Sun ldquoProvincial carbonintensity abatement potential estimation in China a PSO-GA-optimized multi-factor environmental learning curvemethodrdquo Energy Policy vol 77 pp 46ndash55 2015

[20] U-Y Huh and S-R Chang ldquoAG 2 continuous path-smoothing algorithm using modified quadratic polynomialinterpolationrdquo International Journal of Advanced RoboticSystems vol 11 no 25 p 2 2014

[21] B Parmentier B J McGill A M Wilson et al ldquoUsing multi-timescale methods and satellite-derived land surface tem-perature for the interpolation of daily maximum air tem-perature in Oregonrdquo International Journal of Climatologyvol 35 no 13 pp 3862ndash3878 2015

[22] W Kang S Yu S Ko and J Paik ldquoMultisensor super res-olution using directionally-adaptive regularization for UAVimagesrdquo Sensors vol 15 no 5 pp 12053ndash12079 2015

[23] M Gulum M Kadir Yesilyurt and A Bilgin ldquoe perfor-mance assessment of cubic spline interpolation and responsesurface methodology in the mathematical modeling to opti-mize biodiesel production from waste cooking oilrdquo Fuelvol 255 p 115778

[24] W Deng R Yao H Zhao X Yang and G Li ldquoA novelintelligent diagnosis method using optimal LS-SVM withimproved PSO algorithmrdquo Soft Computing vol 23 no 7pp 2445ndash2462 2019

[25] Y Gao ldquoA hybrid method for mobile agent moving trajectoryscheduling using ACO and PSO in WSNsrdquo Sensors vol 19no 3 2019

[26] J Wang ldquoA PSO based energy efficient coverage controlalgorithm for wireless sensor networksrdquoComputers Materialsand Continua vol 56 no 3 pp 433ndash446 2018

[27] C Sahu P B Kumar and D R Parhi ldquoAn intelligent pathplanning approach for humanoid robots using adaptiveparticle swarm optimizationrdquo International Journal on Ar-tificial Intelligence Tools vol 27 no 5 2018

[28] B Jiao Z Lian and X Gu ldquoA dynamic inertia weight particleswarm optimization algorithmrdquo Chaos Solitons amp Fractalsvol 37 no 3 pp 698ndash705 2008

[29] P Melin F Olivas O Castillo F Valdez J Soria andM Valdez ldquoOptimal design of fuzzy classification systemsusing PSO with dynamic parameter adaptation through fuzzylogicrdquo Expert Systems with Applications vol 40 no 8pp 3196ndash3206 2013

[30] T d F Araujo and W Uturbey ldquoPerformance assessment ofPSO DE and hybrid PSO-DE algorithms when applied to thedispatch of generation and demandrdquo International Journal ofElectrical Power amp Energy Systems vol 47 pp 205ndash217 2013

[31] T Y Abdalla A A Abed and A A Ahmed ldquoMobile robotnavigation using PSO-optimized fuzzy artificial potential fieldwith fuzzy controlrdquo Journal of Intelligent amp Fuzzy Systemsvol 32 no 6 pp 3893ndash3908 2017

[32] A Benvidi S Abbasi S Gharaghani M Dehghan Tezerjaniand S Masoum ldquoSpectrophotometric determination ofsynthetic colorants using PSO-GA-ANNrdquo Food Chemistryvol 220 pp 377ndash384 2017

[33] H Wang H Sun C Li S Rahnamayan and J-S PanldquoDiversity enhanced particle swarm optimization withneighborhood searchrdquo Information Sciences vol 223pp 119ndash135 2013

[34] W Ding C T Lin M Prasad Z Cao and J Wang ldquoAlayered-coevolution-based attribute-boosted reduction usingadaptive quantum-behavior PSO and its consistent segmen-tation for neonates brain tissuerdquo IEEE Transactions on FuzzySystems vol 26 no 3 pp 1177ndash1191 2017

[35] X Kong L Gao H Ouyang and S Li ldquoSolving the re-dundancy allocation problem with multiple strategy choicesusing a new simplified particle swarm optimizationrdquo Reli-ability Engineering amp System Safety vol 144 pp 147ndash1582015

[36] A A Nagra F Han Q-H Ling and S Mehta ldquoAn improvedhybrid method combining gravitational search algorithmwithdynamic multi swarm particle swarm optimizationrdquo IEEEAccess vol 7 pp 50388ndash50399 2019

[37] F Q Lu M Huang W K Ching X W Wang and X L SunldquoMulti-swarm particle swarm optimization based risk man-agement model for virtual enterpriserdquo in Proceedings of thefirst ACMSIGEVO Summit on Genetic and EvolutionaryComputation pp 387ndash392 Shanghai China July 2019

[38] G I Sayed G Khoriba and M H Haggag ldquoA novel chaoticsalp swarm algorithm for global optimization and featureselectionrdquo Applied Intelligence vol 48 no 10 pp 3462ndash34812018

[39] S Wang Y Zhang G Ji J Yang J Wu and L Wei ldquoFruitclassification by wavelet-entropy and feedforward neuralnetwork trained by fitness-scaled chaotic abc and biogeog-raphy-based optimizationrdquo Entropy vol 17 no 8pp 5711ndash5728 2015

[40] M Ghasemi S Ghavidel J Aghaei M Gitizadeh andH Falah ldquoApplication of chaos-based chaotic invasive weed


optimization techniques for environmental opf problems inthe power systemrdquo Chaos Solitons amp Fractals vol 69 no 69pp 271ndash284 2014

[41] H L Bian ldquoStudy on chaotic time series prediction algorithmfor kent mapping based on particle swarm optimizationrdquoApplied Mechanics and Materials vol 511-512 pp 941ndash9442014

[42] J Cao ldquoA novel fixed point feedback approach studying thedynamical behaviors of standard logistic maprdquo InternationalJournal of Bifurcation and Chaos vol 29 no 1 2019

[43] E Farri and P Ayubi ldquoA blind and robust video water-marking based on iwt and new 3d generalized chaotic sinemaprdquo Nonlinear Dynamics vol 93 no 4 pp 1875ndash18972018

[44] X Jiang and S Li ldquoBAS beetle antennae search algorithm foroptimization problemsrdquo 2017 httparxivorgabs171010724

[45] R Jensi and G W Jiji ldquoAn enhanced particle swarm opti-mization with levy flight for global optimizationrdquo Applied SoftComputing vol 43 pp 248ndash261 2016

[46] M R Tahir Q X Tran and M S Nikulin ldquoComparison ofhypertabastic survival model with other unimodal hazard ratefunctions using a goodness-of-fit testrdquo Statistics in Medicinevol 36 no 12 pp 1936ndash1945 2017

[47] M Parashar ldquoOptimization of benchmark functions using Anature inspired bird Swarm algorithmrdquo in Proceedings of theInternational Conference on Computational Intelligence ampCommunication Technology pp 9-10 Ghaziabad IndiaFebruary 2017

[48] B K Hassani ldquoDistortion risk measures or the transforma-tion of unimodal distributions into multimodal functionsrdquoDocuments De Travail Du Centre Deconomie De La Sorbonnevol 211 no 5 pp 71ndash88 2015

[49] B Y Qu J J Liang Z Y Wang Q Chen andP N Suganthan ldquoNovel benchmark functions for continuousmultimodal optimization with comparative resultsrdquo Swarmand Evolutionary Computation vol 26 pp 23ndash34 2016

[50] R Moghdani and K Salimifard ldquoVolleyball premier leaguealgorithmrdquo Applied Soft Computing vol 64 pp 161ndash1852018

[51] S Reddy ldquoSolution to unit commitment in power systemoperation planning using binary coded modified moth flameoptimization algorithm (BMMFOA) a flame selection basedcomputational techniquerdquo Journal of Computational Sciencevol 25 pp 298ndash317 2015


population diversity erefore in the robot path planningapplication GA is integrated with other algorithms such asartificial potential field method artificial bee populationalgorithm etc to achieve better results [15] However themain disadvantage of improved genetic algorithms in therobot path planning field is that it takes a long time

e Alowast algorithm is an effective and direct method tosearch path in a static environment However when theAlowast algorithm was used to search in the neighborhood of thecurrent node the path beyond the threshold length isabandoned due to the local path length limitation resultingin the final path being not optimal Also if there are notenough sample points to build the graph the search successrate may be low Nevertheless increasing the number ofsampling points increases costs Hence there are manyimproved Alowast algorithms to improve the optimal range ofrobot paths and search success rates [16] ese improvedAlowast algorithms are very inefficient in dynamic environmentsBecause when the environment changes such as new bar-riers the Alowast algorithm and the improved Alowast algorithmsmust research from the current node to the end of the pathHence if the environment changes and the robotrsquos currentposition is still quite a long distance from the final targetthen the Alowast algorithm and the improved Alowast algorithms willtake a lot of time to redraw its path

e ABC algorithm is based on a simulation of theforaging behavior of the bee colony [11] ABC is easy toimplement ABC also has some drawbacks like early con-vergence stagnation and slow convergence e ABCperforms very well while exploring the feasible search spacebut it shows poor performance for exploitation BesidesABC is first applied to solve the problems of function op-timization while the path planning of a mobile robot is acombinatorial optimization problem erefore there aresome difficulties in algorithm construction

SA is a stochastic global optimization algorithm simu-lated by Kirkpatrick e convergence probability of SAalgorithm is 1 [12] is characteristic can guarantee theachievement of global optimization when it is used for robotglobal path planning while it has the disadvantages of slowconvergence speed e ACO is a random search optimi-zation algorithm with characteristics of positive feedbackand parallel computing which applies to various problemssuch as traveling salesman problem [17] quadratic pro-gramming problem [18] and production scheduling prob-lem [19] However the optimal path of ant colony algorithmplanning is time-consuming

Traditional robot path planning attempts to find anoptimal path which may contain some sharp turns andpolygonal lines However this algorithm is not flexiblebecause it adds extra workload by moving the robot alongthe sharp turn by stopping rotating or restarting evenswitching between different modes erefore traditionalrobot path planning is time-consuming and labor-intensivefor specific tasks requiring smooth motion

Spline interpolation curve is a smooth curve passingthrough a series of given points Quadratic spline interpo-lation is a quadratic polynomial interpolation When thecoefficient of the highest degree term is 0 the quadratic

spline interpolation curve is a straight line [20] ereforethe quadratic interpolation is not applicable However cubicspline interpolation has the convergence property of the firstand second derivatives When the highest term coefficient ofcubic function is 0 the cubic spline interpolation curve isstill a curve and the interpolation effect should be better Inaddition compared with high-order interpolation cubicspline interpolation has the advantages of simple calculationand good stability

Cubic spline interpolation has been widely applied Forexample cubic spline interpolation was used to control thequality of air temperature [21] By interpolating the tem-perature observations for each altitude segment suspiciousvalues in the temperature observation data can be moreeffectively marked Also cubic spline interpolation was usedto simulate the trajectory of the UAV [22] and study the pathsmoothness of the manipulator In [23] the cubic splineinterpolation was applied in the temperature compensationaspect of alcohol concentration measurement results

In this paper the moving robotrsquos path can be drawn as aseries of line segments that join the path nodes e cubicspline function was used to interpolate on the path of thestarting point control points and target point us a fullpath that was formed by connecting all interpolation pointswas obtained e control pointsrsquo position determines thelength of the path the number of which determines thenumber of spline curves and the maximum turnings ofthe patherefore these control points are used to optimizethe path of mobile robots Heuristic optimization algorithmsare an alternative to optimize control points

Particle swarm optimization (PSO) is a computationaltechnology Like other evolutionary algorithms PSO ach-ieves the search for optimal spatial solutions through in-dividual collaboration and competition However comparedwith evolutionary algorithms PSO is simple and easy toimplement Moreover PSO needs to adjust relatively fewparameters which is convenient and practical PSO has beenwidely used in many applications For example the PSOalgorithm is applied to deal with mathematical problems[24] which shows excellent performance in solving someclassical function optimization problems and even somenonlinear functions Also the PSO algorithm is introducedinto the fault diagnosis to propose a novel intelligent di-agnosis method which is applied to diagnose the faults of themotor bearing [25] en PSO is also used to solve theproblem of optimal deployment of wireless sensor network(WSN) nodes and to solve many problems caused by theoptimized deployment of static nodes [26]

However in some cases particle swarm optimizationcannot find a global optimal solution due to particle de-pletion Moreover the search strategy of the particle swarmalgorithm is mainly based on a random walk so it cannotalways successfully find the optimal solution To date somevariants of PSO have been presented to improve the originalversion PSO Most of the current existing PSOs can beroughly divided into three categories parameter selectionhybrid versions and topology structure respectively [27]

e proper selection of control parameters such asinertia weight and acceleration coefficient can significantly















2 Background





xi(t + 1) xi(t) + vi(t + 1) (2)







+ di x minus xi( 11138573 (4)


f x0( 1113857 y0 f xn+1( 1113857 yn+1 (5)

fminus f+ xi( 1113857 yi i 1 2 n (6)



prime xi( 1113857 i 1 2 n (7)



Prime xi( 1113857 i 1 2 n (8)




minP(t) 1113944d

i1

xi+1 minus xi( 11138572

+ yi+1 minus yi( 11138572

1113969

(9)



gbest

pbest

vi (t + 1)

xi (t + 1)



xi (t )

vi (t )





dsafe


2+ yi minus Ok

y1113872 11138732

minus cOk

1113970

(10)

where Okx and Ok





Min P(t)

stP(t) isin C2


(12)







xk+1 μ 786xk minus 2331x2k + 2875x

3k minus 133x

4k1113872 1113873 (13)



xk+1 a

4sin πxk( 1113857 (14)



xk+1

xk

a 0lt xlt a

1 minus xk

1 minus a altxlt 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)



xk+1 axk 1 minus xk( 1113857 (16)


iteration is









dir rands(n 1) (18)

dir dir

norm(dir) (19)




d0 step

c

(20)





1113888 1113889

1 1+ 10lowastkkmax111385711138571113857((((22)


05

10152025303540

(a)

0

10

20

30

40

50

(b)

0

10

20

30

40

50

(c)

05

101520253035404550

(d)




fleft f(xl)

fright f(xr)(23)




which is


(25)






2lowast cos

πlowast k

kmax1113888 1113889 +


2 (27)


C1(k) cosπlowast k

kmax1113888 1113889 + a (28)


kmax1113888 1113889 + a (29)









Start




Update r1 and r2




Produce path

Output result

End

Boundary processing


Update the path

Y

N

Updated parameter

Logistic map

Sine map

Singer map

Kent map

Choose a map



04

05

06

07

08

09 W

val

ue

A

B

C

D


(a)


05

1

15

2

25

Val

ue

E

F

G

K

L

M

N

O

P

Q



(b)




6 Experiments

























ni1 x2

i 30 [minus 30 30] 0F2(x) 1113936

ni1 |xi| + 1113937

ni1 |xi| 30 [minus 30 30] 0

F3(xi) 1113936ni1[x2


(1n) 1113936

nij x2

i

1113969) minus exp((1n) 1113936

ni1 cos(2πi) + 20 + e 30 [minus 32 32] 0



1 + (13)x61 + x1x2 minus 4x2

2 + 4x41 2 [minus 5 5] 0



005

1

15

2

10050

0ndash50

ndash100


0020406081121416182

x2 x1

times104

times104

(a)

020406080

100



10

20

30

40

50

60

70

80

x2 x1

(b)

060

100

40

200

8020 60

300

400

400

200

500


50100150200250300350400450

x2 x1

(c)






103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(a)




103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(b)




25

20

15

10

5

0

Fitn

ess v

alue

(c)




101

100

Fitn

ess v

alue

(d)




102

101

100

Fitn

ess v

alue

(e)




7

6

5

4

3

2

1

0

Fitn

ess v

alue

(f )







656

555

454

353

252

151 2 3 4

n = 15 6 7

(a)

656

555

454

353

252

151 2 3 4 5 6 7

n = 2

(b)

656

555

454

353

252

151 2 3 4 5 6 7

n = 3

(c)

656

555

454

353

252

151 2 3 4 5 6 7

n = 4

(d)

656

555

454

353

252

151 2 3 4 5 6 7

n = 5

(e)

656

555

454

353

252

151 2 3 4 5 6 7

n = 6

(f )

656

555

454

353

252

15

1 2 3 4 5 6 7n = 7

(g)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 8

(h)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 9

(i)






80

70

60

50

40

30

20

10

00 20 40 60 80 100


Fitn

ess v

alue

n = 1n = 2n = 3n = 4n = 5

n = 6n = 7n = 8n = 9


0 5 10 15 200

5

10

15

20

PSOCAPSOGA

DESA

(a)


25

30

35

40

45

50

55Fi

tnes

s val

ue

PSOCAPSOGA

DESA

(b)




Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


DE 328655 338655 350157 123581 11001 329GA 314532 324532 336302 56875 10895 50SA 322260 359860 341927 261047 09834 396PSO 300592 320592 318083 50907 08744 50CAPSO 297213 299821 311723 47042 07255 44













Mean fitnessvalue

Worst fitnessvalue


Standarddeviation





Mean fitnessvalue

Worst fitnessvalue


Standarddeviation




0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)






























































2 Background





xi(t + 1) xi(t) + vi(t + 1) (2)







+ di x minus xi( 11138573 (4)


f x0( 1113857 y0 f xn+1( 1113857 yn+1 (5)

fminus f+ xi( 1113857 yi i 1 2 n (6)



prime xi( 1113857 i 1 2 n (7)



Prime xi( 1113857 i 1 2 n (8)




minP(t) 1113944d

i1

xi+1 minus xi( 11138572

+ yi+1 minus yi( 11138572

1113969

(9)



gbest

pbest

vi (t + 1)

xi (t + 1)



xi (t )

vi (t )





dsafe


2+ yi minus Ok

y1113872 11138732

minus cOk

1113970

(10)

where Okx and Ok





Min P(t)

stP(t) isin C2


(12)







xk+1 μ 786xk minus 2331x2k + 2875x

3k minus 133x

4k1113872 1113873 (13)



xk+1 a

4sin πxk( 1113857 (14)



xk+1

xk

a 0lt xlt a

1 minus xk

1 minus a altxlt 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)



xk+1 axk 1 minus xk( 1113857 (16)


iteration is









dir rands(n 1) (18)

dir dir

norm(dir) (19)




d0 step

c

(20)





1113888 1113889

1 1+ 10lowastkkmax111385711138571113857((((22)


05

10152025303540

(a)

0

10

20

30

40

50

(b)

0

10

20

30

40

50

(c)

05

101520253035404550

(d)




fleft f(xl)

fright f(xr)(23)




which is


(25)






2lowast cos

πlowast k

kmax1113888 1113889 +


2 (27)


C1(k) cosπlowast k

kmax1113888 1113889 + a (28)


kmax1113888 1113889 + a (29)









Start




Update r1 and r2




Produce path

Output result

End

Boundary processing


Update the path

Y

N

Updated parameter

Logistic map

Sine map

Singer map

Kent map

Choose a map



04

05

06

07

08

09 W

val

ue

A

B

C

D


(a)


05

1

15

2

25

Val

ue

E

F

G

K

L

M

N

O

P

Q



(b)




6 Experiments

























ni1 x2

i 30 [minus 30 30] 0F2(x) 1113936

ni1 |xi| + 1113937

ni1 |xi| 30 [minus 30 30] 0

F3(xi) 1113936ni1[x2


(1n) 1113936

nij x2

i

1113969) minus exp((1n) 1113936

ni1 cos(2πi) + 20 + e 30 [minus 32 32] 0



1 + (13)x61 + x1x2 minus 4x2

2 + 4x41 2 [minus 5 5] 0



005

1

15

2

10050

0ndash50

ndash100


0020406081121416182

x2 x1

times104

times104

(a)

020406080

100



10

20

30

40

50

60

70

80

x2 x1

(b)

060

100

40

200

8020 60

300

400

400

200

500


50100150200250300350400450

x2 x1

(c)






103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(a)




103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(b)




25

20

15

10

5

0

Fitn

ess v

alue

(c)




101

100

Fitn

ess v

alue

(d)




102

101

100

Fitn

ess v

alue

(e)




7

6

5

4

3

2

1

0

Fitn

ess v

alue

(f )







656

555

454

353

252

151 2 3 4

n = 15 6 7

(a)

656

555

454

353

252

151 2 3 4 5 6 7

n = 2

(b)

656

555

454

353

252

151 2 3 4 5 6 7

n = 3

(c)

656

555

454

353

252

151 2 3 4 5 6 7

n = 4

(d)

656

555

454

353

252

151 2 3 4 5 6 7

n = 5

(e)

656

555

454

353

252

151 2 3 4 5 6 7

n = 6

(f )

656

555

454

353

252

15

1 2 3 4 5 6 7n = 7

(g)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 8

(h)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 9

(i)






80

70

60

50

40

30

20

10

00 20 40 60 80 100


Fitn

ess v

alue

n = 1n = 2n = 3n = 4n = 5

n = 6n = 7n = 8n = 9


0 5 10 15 200

5

10

15

20

PSOCAPSOGA

DESA

(a)


25

30

35

40

45

50

55Fi

tnes

s val

ue

PSOCAPSOGA

DESA

(b)




Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


DE 328655 338655 350157 123581 11001 329GA 314532 324532 336302 56875 10895 50SA 322260 359860 341927 261047 09834 396PSO 300592 320592 318083 50907 08744 50CAPSO 297213 299821 311723 47042 07255 44













Mean fitnessvalue

Worst fitnessvalue


Standarddeviation





Mean fitnessvalue

Worst fitnessvalue


Standarddeviation




0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)



















































dsafe


2+ yi minus Ok

y1113872 11138732

minus cOk

1113970

(10)

where Okx and Ok





Min P(t)

stP(t) isin C2


(12)







xk+1 μ 786xk minus 2331x2k + 2875x

3k minus 133x

4k1113872 1113873 (13)



xk+1 a

4sin πxk( 1113857 (14)



xk+1

xk

a 0lt xlt a

1 minus xk

1 minus a altxlt 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)



xk+1 axk 1 minus xk( 1113857 (16)


iteration is









dir rands(n 1) (18)

dir dir

norm(dir) (19)




d0 step

c

(20)





1113888 1113889

1 1+ 10lowastkkmax111385711138571113857((((22)


05

10152025303540

(a)

0

10

20

30

40

50

(b)

0

10

20

30

40

50

(c)

05

101520253035404550

(d)




fleft f(xl)

fright f(xr)(23)




which is


(25)






2lowast cos

πlowast k

kmax1113888 1113889 +


2 (27)


C1(k) cosπlowast k

kmax1113888 1113889 + a (28)


kmax1113888 1113889 + a (29)









Start




Update r1 and r2




Produce path

Output result

End

Boundary processing


Update the path

Y

N

Updated parameter

Logistic map

Sine map

Singer map

Kent map

Choose a map



04

05

06

07

08

09 W

val

ue

A

B

C

D


(a)


05

1

15

2

25

Val

ue

E

F

G

K

L

M

N

O

P

Q



(b)




6 Experiments

























ni1 x2

i 30 [minus 30 30] 0F2(x) 1113936

ni1 |xi| + 1113937

ni1 |xi| 30 [minus 30 30] 0

F3(xi) 1113936ni1[x2


(1n) 1113936

nij x2

i

1113969) minus exp((1n) 1113936

ni1 cos(2πi) + 20 + e 30 [minus 32 32] 0



1 + (13)x61 + x1x2 minus 4x2

2 + 4x41 2 [minus 5 5] 0



005

1

15

2

10050

0ndash50

ndash100


0020406081121416182

x2 x1

times104

times104

(a)

020406080

100



10

20

30

40

50

60

70

80

x2 x1

(b)

060

100

40

200

8020 60

300

400

400

200

500


50100150200250300350400450

x2 x1

(c)






103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(a)




103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(b)




25

20

15

10

5

0

Fitn

ess v

alue

(c)




101

100

Fitn

ess v

alue

(d)




102

101

100

Fitn

ess v

alue

(e)




7

6

5

4

3

2

1

0

Fitn

ess v

alue

(f )







656

555

454

353

252

151 2 3 4

n = 15 6 7

(a)

656

555

454

353

252

151 2 3 4 5 6 7

n = 2

(b)

656

555

454

353

252

151 2 3 4 5 6 7

n = 3

(c)

656

555

454

353

252

151 2 3 4 5 6 7

n = 4

(d)

656

555

454

353

252

151 2 3 4 5 6 7

n = 5

(e)

656

555

454

353

252

151 2 3 4 5 6 7

n = 6

(f )

656

555

454

353

252

15

1 2 3 4 5 6 7n = 7

(g)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 8

(h)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 9

(i)






80

70

60

50

40

30

20

10

00 20 40 60 80 100


Fitn

ess v

alue

n = 1n = 2n = 3n = 4n = 5

n = 6n = 7n = 8n = 9


0 5 10 15 200

5

10

15

20

PSOCAPSOGA

DESA

(a)


25

30

35

40

45

50

55Fi

tnes

s val

ue

PSOCAPSOGA

DESA

(b)




Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


DE 328655 338655 350157 123581 11001 329GA 314532 324532 336302 56875 10895 50SA 322260 359860 341927 261047 09834 396PSO 300592 320592 318083 50907 08744 50CAPSO 297213 299821 311723 47042 07255 44













Mean fitnessvalue

Worst fitnessvalue


Standarddeviation





Mean fitnessvalue

Worst fitnessvalue


Standarddeviation




0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)





















































dir rands(n 1) (18)

dir dir

norm(dir) (19)




d0 step

c

(20)





1113888 1113889

1 1+ 10lowastkkmax111385711138571113857((((22)


05

10152025303540

(a)

0

10

20

30

40

50

(b)

0

10

20

30

40

50

(c)

05

101520253035404550

(d)




fleft f(xl)

fright f(xr)(23)




which is


(25)






2lowast cos

πlowast k

kmax1113888 1113889 +


2 (27)


C1(k) cosπlowast k

kmax1113888 1113889 + a (28)


kmax1113888 1113889 + a (29)









Start




Update r1 and r2




Produce path

Output result

End

Boundary processing


Update the path

Y

N

Updated parameter

Logistic map

Sine map

Singer map

Kent map

Choose a map



04

05

06

07

08

09 W

val

ue

A

B

C

D


(a)


05

1

15

2

25

Val

ue

E

F

G

K

L

M

N

O

P

Q



(b)




6 Experiments

























ni1 x2

i 30 [minus 30 30] 0F2(x) 1113936

ni1 |xi| + 1113937

ni1 |xi| 30 [minus 30 30] 0

F3(xi) 1113936ni1[x2


(1n) 1113936

nij x2

i

1113969) minus exp((1n) 1113936

ni1 cos(2πi) + 20 + e 30 [minus 32 32] 0



1 + (13)x61 + x1x2 minus 4x2

2 + 4x41 2 [minus 5 5] 0



005

1

15

2

10050

0ndash50

ndash100


0020406081121416182

x2 x1

times104

times104

(a)

020406080

100



10

20

30

40

50

60

70

80

x2 x1

(b)

060

100

40

200

8020 60

300

400

400

200

500


50100150200250300350400450

x2 x1

(c)






103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(a)




103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(b)




25

20

15

10

5

0

Fitn

ess v

alue

(c)




101

100

Fitn

ess v

alue

(d)




102

101

100

Fitn

ess v

alue

(e)




7

6

5

4

3

2

1

0

Fitn

ess v

alue

(f )







656

555

454

353

252

151 2 3 4

n = 15 6 7

(a)

656

555

454

353

252

151 2 3 4 5 6 7

n = 2

(b)

656

555

454

353

252

151 2 3 4 5 6 7

n = 3

(c)

656

555

454

353

252

151 2 3 4 5 6 7

n = 4

(d)

656

555

454

353

252

151 2 3 4 5 6 7

n = 5

(e)

656

555

454

353

252

151 2 3 4 5 6 7

n = 6

(f )

656

555

454

353

252

15

1 2 3 4 5 6 7n = 7

(g)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 8

(h)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 9

(i)






80

70

60

50

40

30

20

10

00 20 40 60 80 100


Fitn

ess v

alue

n = 1n = 2n = 3n = 4n = 5

n = 6n = 7n = 8n = 9


0 5 10 15 200

5

10

15

20

PSOCAPSOGA

DESA

(a)


25

30

35

40

45

50

55Fi

tnes

s val

ue

PSOCAPSOGA

DESA

(b)




Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


DE 328655 338655 350157 123581 11001 329GA 314532 324532 336302 56875 10895 50SA 322260 359860 341927 261047 09834 396PSO 300592 320592 318083 50907 08744 50CAPSO 297213 299821 311723 47042 07255 44













Mean fitnessvalue

Worst fitnessvalue


Standarddeviation





Mean fitnessvalue

Worst fitnessvalue


Standarddeviation




0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)


















































fleft f(xl)

fright f(xr)(23)




which is


(25)






2lowast cos

πlowast k

kmax1113888 1113889 +


2 (27)


C1(k) cosπlowast k

kmax1113888 1113889 + a (28)


kmax1113888 1113889 + a (29)









Start




Update r1 and r2




Produce path

Output result

End

Boundary processing


Update the path

Y

N

Updated parameter

Logistic map

Sine map

Singer map

Kent map

Choose a map



04

05

06

07

08

09 W

val

ue

A

B

C

D


(a)


05

1

15

2

25

Val

ue

E

F

G

K

L

M

N

O

P

Q



(b)




6 Experiments

























ni1 x2

i 30 [minus 30 30] 0F2(x) 1113936

ni1 |xi| + 1113937

ni1 |xi| 30 [minus 30 30] 0

F3(xi) 1113936ni1[x2


(1n) 1113936

nij x2

i

1113969) minus exp((1n) 1113936

ni1 cos(2πi) + 20 + e 30 [minus 32 32] 0



1 + (13)x61 + x1x2 minus 4x2

2 + 4x41 2 [minus 5 5] 0



005

1

15

2

10050

0ndash50

ndash100


0020406081121416182

x2 x1

times104

times104

(a)

020406080

100



10

20

30

40

50

60

70

80

x2 x1

(b)

060

100

40

200

8020 60

300

400

400

200

500


50100150200250300350400450

x2 x1

(c)






103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(a)




103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(b)




25

20

15

10

5

0

Fitn

ess v

alue

(c)




101

100

Fitn

ess v

alue

(d)




102

101

100

Fitn

ess v

alue

(e)




7

6

5

4

3

2

1

0

Fitn

ess v

alue

(f )







656

555

454

353

252

151 2 3 4

n = 15 6 7

(a)

656

555

454

353

252

151 2 3 4 5 6 7

n = 2

(b)

656

555

454

353

252

151 2 3 4 5 6 7

n = 3

(c)

656

555

454

353

252

151 2 3 4 5 6 7

n = 4

(d)

656

555

454

353

252

151 2 3 4 5 6 7

n = 5

(e)

656

555

454

353

252

151 2 3 4 5 6 7

n = 6

(f )

656

555

454

353

252

15

1 2 3 4 5 6 7n = 7

(g)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 8

(h)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 9

(i)






80

70

60

50

40

30

20

10

00 20 40 60 80 100


Fitn

ess v

alue

n = 1n = 2n = 3n = 4n = 5

n = 6n = 7n = 8n = 9


0 5 10 15 200

5

10

15

20

PSOCAPSOGA

DESA

(a)


25

30

35

40

45

50

55Fi

tnes

s val

ue

PSOCAPSOGA

DESA

(b)




Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


DE 328655 338655 350157 123581 11001 329GA 314532 324532 336302 56875 10895 50SA 322260 359860 341927 261047 09834 396PSO 300592 320592 318083 50907 08744 50CAPSO 297213 299821 311723 47042 07255 44













Mean fitnessvalue

Worst fitnessvalue


Standarddeviation





Mean fitnessvalue

Worst fitnessvalue


Standarddeviation




0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)

















































Start




Update r1 and r2




Produce path

Output result

End

Boundary processing


Update the path

Y

N

Updated parameter

Logistic map

Sine map

Singer map

Kent map

Choose a map



04

05

06

07

08

09 W

val

ue

A

B

C

D


(a)


05

1

15

2

25

Val

ue

E

F

G

K

L

M

N

O

P

Q



(b)




6 Experiments

























ni1 x2

i 30 [minus 30 30] 0F2(x) 1113936

ni1 |xi| + 1113937

ni1 |xi| 30 [minus 30 30] 0

F3(xi) 1113936ni1[x2


(1n) 1113936

nij x2

i

1113969) minus exp((1n) 1113936

ni1 cos(2πi) + 20 + e 30 [minus 32 32] 0



1 + (13)x61 + x1x2 minus 4x2

2 + 4x41 2 [minus 5 5] 0



005

1

15

2

10050

0ndash50

ndash100


0020406081121416182

x2 x1

times104

times104

(a)

020406080

100



10

20

30

40

50

60

70

80

x2 x1

(b)

060

100

40

200

8020 60

300

400

400

200

500


50100150200250300350400450

x2 x1

(c)






103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(a)




103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(b)




25

20

15

10

5

0

Fitn

ess v

alue

(c)




101

100

Fitn

ess v

alue

(d)




102

101

100

Fitn

ess v

alue

(e)




7

6

5

4

3

2

1

0

Fitn

ess v

alue

(f )







656

555

454

353

252

151 2 3 4

n = 15 6 7

(a)

656

555

454

353

252

151 2 3 4 5 6 7

n = 2

(b)

656

555

454

353

252

151 2 3 4 5 6 7

n = 3

(c)

656

555

454

353

252

151 2 3 4 5 6 7

n = 4

(d)

656

555

454

353

252

151 2 3 4 5 6 7

n = 5

(e)

656

555

454

353

252

151 2 3 4 5 6 7

n = 6

(f )

656

555

454

353

252

15

1 2 3 4 5 6 7n = 7

(g)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 8

(h)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 9

(i)






80

70

60

50

40

30

20

10

00 20 40 60 80 100


Fitn

ess v

alue

n = 1n = 2n = 3n = 4n = 5

n = 6n = 7n = 8n = 9


0 5 10 15 200

5

10

15

20

PSOCAPSOGA

DESA

(a)


25

30

35

40

45

50

55Fi

tnes

s val

ue

PSOCAPSOGA

DESA

(b)




Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


DE 328655 338655 350157 123581 11001 329GA 314532 324532 336302 56875 10895 50SA 322260 359860 341927 261047 09834 396PSO 300592 320592 318083 50907 08744 50CAPSO 297213 299821 311723 47042 07255 44













Mean fitnessvalue

Worst fitnessvalue


Standarddeviation





Mean fitnessvalue

Worst fitnessvalue


Standarddeviation




0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)


















































6 Experiments

























ni1 x2

i 30 [minus 30 30] 0F2(x) 1113936

ni1 |xi| + 1113937

ni1 |xi| 30 [minus 30 30] 0

F3(xi) 1113936ni1[x2


(1n) 1113936

nij x2

i

1113969) minus exp((1n) 1113936

ni1 cos(2πi) + 20 + e 30 [minus 32 32] 0



1 + (13)x61 + x1x2 minus 4x2

2 + 4x41 2 [minus 5 5] 0



005

1

15

2

10050

0ndash50

ndash100


0020406081121416182

x2 x1

times104

times104

(a)

020406080

100



10

20

30

40

50

60

70

80

x2 x1

(b)

060

100

40

200

8020 60

300

400

400

200

500


50100150200250300350400450

x2 x1

(c)






103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(a)




103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(b)




25

20

15

10

5

0

Fitn

ess v

alue

(c)




101

100

Fitn

ess v

alue

(d)




102

101

100

Fitn

ess v

alue

(e)




7

6

5

4

3

2

1

0

Fitn

ess v

alue

(f )







656

555

454

353

252

151 2 3 4

n = 15 6 7

(a)

656

555

454

353

252

151 2 3 4 5 6 7

n = 2

(b)

656

555

454

353

252

151 2 3 4 5 6 7

n = 3

(c)

656

555

454

353

252

151 2 3 4 5 6 7

n = 4

(d)

656

555

454

353

252

151 2 3 4 5 6 7

n = 5

(e)

656

555

454

353

252

151 2 3 4 5 6 7

n = 6

(f )

656

555

454

353

252

15

1 2 3 4 5 6 7n = 7

(g)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 8

(h)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 9

(i)






80

70

60

50

40

30

20

10

00 20 40 60 80 100


Fitn

ess v

alue

n = 1n = 2n = 3n = 4n = 5

n = 6n = 7n = 8n = 9


0 5 10 15 200

5

10

15

20

PSOCAPSOGA

DESA

(a)


25

30

35

40

45

50

55Fi

tnes

s val

ue

PSOCAPSOGA

DESA

(b)




Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


DE 328655 338655 350157 123581 11001 329GA 314532 324532 336302 56875 10895 50SA 322260 359860 341927 261047 09834 396PSO 300592 320592 318083 50907 08744 50CAPSO 297213 299821 311723 47042 07255 44













Mean fitnessvalue

Worst fitnessvalue


Standarddeviation





Mean fitnessvalue

Worst fitnessvalue


Standarddeviation




0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)


























































ni1 x2

i 30 [minus 30 30] 0F2(x) 1113936

ni1 |xi| + 1113937

ni1 |xi| 30 [minus 30 30] 0

F3(xi) 1113936ni1[x2


(1n) 1113936

nij x2

i

1113969) minus exp((1n) 1113936

ni1 cos(2πi) + 20 + e 30 [minus 32 32] 0



1 + (13)x61 + x1x2 minus 4x2

2 + 4x41 2 [minus 5 5] 0



005

1

15

2

10050

0ndash50

ndash100


0020406081121416182

x2 x1

times104

times104

(a)

020406080

100



10

20

30

40

50

60

70

80

x2 x1

(b)

060

100

40

200

8020 60

300

400

400

200

500


50100150200250300350400450

x2 x1

(c)






103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(a)




103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(b)




25

20

15

10

5

0

Fitn

ess v

alue

(c)




101

100

Fitn

ess v

alue

(d)




102

101

100

Fitn

ess v

alue

(e)




7

6

5

4

3

2

1

0

Fitn

ess v

alue

(f )







656

555

454

353

252

151 2 3 4

n = 15 6 7

(a)

656

555

454

353

252

151 2 3 4 5 6 7

n = 2

(b)

656

555

454

353

252

151 2 3 4 5 6 7

n = 3

(c)

656

555

454

353

252

151 2 3 4 5 6 7

n = 4

(d)

656

555

454

353

252

151 2 3 4 5 6 7

n = 5

(e)

656

555

454

353

252

151 2 3 4 5 6 7

n = 6

(f )

656

555

454

353

252

15

1 2 3 4 5 6 7n = 7

(g)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 8

(h)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 9

(i)






80

70

60

50

40

30

20

10

00 20 40 60 80 100


Fitn

ess v

alue

n = 1n = 2n = 3n = 4n = 5

n = 6n = 7n = 8n = 9


0 5 10 15 200

5

10

15

20

PSOCAPSOGA

DESA

(a)


25

30

35

40

45

50

55Fi

tnes

s val

ue

PSOCAPSOGA

DESA

(b)




Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


DE 328655 338655 350157 123581 11001 329GA 314532 324532 336302 56875 10895 50SA 322260 359860 341927 261047 09834 396PSO 300592 320592 318083 50907 08744 50CAPSO 297213 299821 311723 47042 07255 44













Mean fitnessvalue

Worst fitnessvalue


Standarddeviation





Mean fitnessvalue

Worst fitnessvalue


Standarddeviation




0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)




















































103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(a)




103

102

101

100

10ndash1

10ndash2

Fitn

ess v

alue

(b)




25

20

15

10

5

0

Fitn

ess v

alue

(c)




101

100

Fitn

ess v

alue

(d)




102

101

100

Fitn

ess v

alue

(e)




7

6

5

4

3

2

1

0

Fitn

ess v

alue

(f )







656

555

454

353

252

151 2 3 4

n = 15 6 7

(a)

656

555

454

353

252

151 2 3 4 5 6 7

n = 2

(b)

656

555

454

353

252

151 2 3 4 5 6 7

n = 3

(c)

656

555

454

353

252

151 2 3 4 5 6 7

n = 4

(d)

656

555

454

353

252

151 2 3 4 5 6 7

n = 5

(e)

656

555

454

353

252

151 2 3 4 5 6 7

n = 6

(f )

656

555

454

353

252

15

1 2 3 4 5 6 7n = 7

(g)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 8

(h)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 9

(i)






80

70

60

50

40

30

20

10

00 20 40 60 80 100


Fitn

ess v

alue

n = 1n = 2n = 3n = 4n = 5

n = 6n = 7n = 8n = 9


0 5 10 15 200

5

10

15

20

PSOCAPSOGA

DESA

(a)


25

30

35

40

45

50

55Fi

tnes

s val

ue

PSOCAPSOGA

DESA

(b)




Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


DE 328655 338655 350157 123581 11001 329GA 314532 324532 336302 56875 10895 50SA 322260 359860 341927 261047 09834 396PSO 300592 320592 318083 50907 08744 50CAPSO 297213 299821 311723 47042 07255 44













Mean fitnessvalue

Worst fitnessvalue


Standarddeviation





Mean fitnessvalue

Worst fitnessvalue


Standarddeviation




0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)





















































656

555

454

353

252

151 2 3 4

n = 15 6 7

(a)

656

555

454

353

252

151 2 3 4 5 6 7

n = 2

(b)

656

555

454

353

252

151 2 3 4 5 6 7

n = 3

(c)

656

555

454

353

252

151 2 3 4 5 6 7

n = 4

(d)

656

555

454

353

252

151 2 3 4 5 6 7

n = 5

(e)

656

555

454

353

252

151 2 3 4 5 6 7

n = 6

(f )

656

555

454

353

252

15

1 2 3 4 5 6 7n = 7

(g)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 8

(h)

656

555

454

353

252

15

1 2 3 4 5 6 7n = 9

(i)






80

70

60

50

40

30

20

10

00 20 40 60 80 100


Fitn

ess v

alue

n = 1n = 2n = 3n = 4n = 5

n = 6n = 7n = 8n = 9


0 5 10 15 200

5

10

15

20

PSOCAPSOGA

DESA

(a)


25

30

35

40

45

50

55Fi

tnes

s val

ue

PSOCAPSOGA

DESA

(b)




Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


DE 328655 338655 350157 123581 11001 329GA 314532 324532 336302 56875 10895 50SA 322260 359860 341927 261047 09834 396PSO 300592 320592 318083 50907 08744 50CAPSO 297213 299821 311723 47042 07255 44













Mean fitnessvalue

Worst fitnessvalue


Standarddeviation





Mean fitnessvalue

Worst fitnessvalue


Standarddeviation




0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)




















































80

70

60

50

40

30

20

10

00 20 40 60 80 100


Fitn

ess v

alue

n = 1n = 2n = 3n = 4n = 5

n = 6n = 7n = 8n = 9


0 5 10 15 200

5

10

15

20

PSOCAPSOGA

DESA

(a)


25

30

35

40

45

50

55Fi

tnes

s val

ue

PSOCAPSOGA

DESA

(b)




Mean fitnessvalue

Worst fitnessvalue


Standarddeviation


DE 328655 338655 350157 123581 11001 329GA 314532 324532 336302 56875 10895 50SA 322260 359860 341927 261047 09834 396PSO 300592 320592 318083 50907 08744 50CAPSO 297213 299821 311723 47042 07255 44













Mean fitnessvalue

Worst fitnessvalue


Standarddeviation





Mean fitnessvalue

Worst fitnessvalue


Standarddeviation




0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)




























































Mean fitnessvalue

Worst fitnessvalue


Standarddeviation





Mean fitnessvalue

Worst fitnessvalue


Standarddeviation




0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

400

300

200

100

0500 6004003002001000

(b)

Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









500

400

300

200

100

0500 6004003002001000

(c)

500

400

300

200

100

0500 6004003002001000

(d)

















































0 100 200 300 400 500 600 700 800 900 1000 1100

0

100

200

300

400

500

600

700

800

900

1000

1100

(a)

2000

3000

4000

5000

6000

700080009000

100001100012000

Fitn

ess v

alue


MPSODEPSOSPSO

IWPSOFAPSOCAPSO

(b)


ndash2 0 2 4 6 8 10 120

2

4

6

8

10

12

(a)

18

20

22

24

26

28

30

32

Fitn

ess f

unct

ion


CAPSOIWPSOMPSO

SPSOFAPSODEPSO

(b)





1000

800

600

400

200

0

10008006004002000

(a)

10008006004002000

1000

800

600

400

200

0

(b)

10008006004002000

1000

800

600

400

200

0

(c)

10008006004002000

1000

800

600

400

200

0

(d)



1000

2000

3000

4000

5000

6000

Fitn

ess v

alue



12

10

8

6

4

2

0121086420

(a)

12

10

8

6

4

2

0121086420

(b)

12

10

8

6

4

2

0121086420

(c)

12

10

8

6

4

2

0121086420

(d)

12

10

8

6

4

2

0121086420

(e)

12

10

8

6

4

2

0121086420

(f )


500

400

300

200

100

0500 6004003002001000

(a)

500

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Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









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Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









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Figure 15 Continued



7 Conclusions


Data Availability




Acknowledgments


References









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7 Conclusions


Data Availability




Acknowledgments


References









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