Research ArticleRobot Path Planning Based on Genetic Algorithm Fused withContinuous Bezier Optimization

Jianwei Ma Yang Liu Shaofei Zang and Lin Wang

School of Information Engineering Henan University of Science and Technology Luoyang 471023 Henan China

Correspondence should be addressed to Yang Liu yangliustuhausteducn

Received 26 September 2019 Revised 16 December 2019 Accepted 24 January 2020 Published 25 February 2020

Academic Editor Rasit Koker

Copyright copy 2020 Jianwei Ma 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

In this study a new method of smooth path planning is proposed based on Bezier curves and is applied to solve the problem ofredundant nodes and peak inflection points in the path planning process of traditional algorithms First genetic operations areused to obtain the control points of the Bezier curve Second a shorter path is selected by an optimization criterion that the lengthof the Bezier curve is determined by the control points Finally a safe distance and adaptive penalty factor are introduced into thefitness function to ensure the safety of the walking process of the robot Numerous experiments are implemented in two differentenvironments and compared with the existing methods It is proved that the proposed method is more effective to generate ashorter smoother and safer path compared with traditional approaches

1 Introduction

Path planning is an important research direction in the fieldof mobile robots and it is one of the main difficulties inresearch on such robots [1]e path planning problem aimsto find the safest and shortest path autonomously withoutcollisions from the start point to the target point under agiven environment with barriers [2 3] Path planning hasbeen widely used in fields such as logistics distributionintelligent transportation and weapons navigation [4ndash6]erefore how to find a fast and effective path has become aresearch issue with a high theoretical significance andpractical value

In recent years the genetic algorithm (GA) has beenwidely applied in mobile robot path planning problemsbecause of its great global optimization ability and implicitparallel computing characteristics [7 8] e GA searchesfor the optimal solution by simulating the natural evolu-tion based on the theoretical models of the genetic in-heritance and variation in Darwinrsquos biological evolution[9 10] Recently some meaningful results have been re-ported for the GA For example a generalized segmen-tation crossover operator was introduced into the GA toimprove the local optimization ability and execution

efficiency of the algorithm [11] Albayrak and Allahverdiintroduced a greedy search algorithm into the GA muta-tion operation and designed a new greedy sunbath mu-tation operator to solve the traveling salesman problem(TSP) [12 13] Furthermore an improved crossover op-erator was proposed [14] in which premature convergencecould be avoided to obtain an optimal path in static en-vironments A robot path planning method was proposedbased on the improved genetic algorithm [15] in which theadaptability of a mobile robot path planning algorithm wasimproved by introducing chromosomes with variablelengths A parallel elite genetic algorithm was proposed tomaintain population diversity avoid premature conver-gence and maintain parallelism with traditional geneticalgorithms [16] However it is noteworthy that there stillexist many problems in the planning process For examplethe spikes and inflection points in the obtained path willmake the robot unable to walk along the planned pathduring the moving process or it will switch frequentlybetween different modes such as stop rotate and restartwhich leads to excessive loss of time and energy etc [6 17]

Recently the Bezier curve has been applied in smoothpath planning (see [6 18ndash27]) As one example a geneticalgorithm was proposed to find the control points of the

HindawiComputational Intelligence and NeuroscienceVolume 2020 Article ID 9813040 10 pageshttpsdoiorg10115520209813040

segmented Bezier curves and thus solve the problem of themobile robot path planning [16] A path planning methodwas proposed based on the Bezier curve to solve the travelingpath in multiagent robot soccer [21] A collision-free cur-vature bounded smooth path planning technique wasproposed to divide the nodes in the piecewise linear pathinto control points [22] Furthermore a Bezier-curve-basedimproved genetic algorithm (MGA) was proposed and usedfor path planning in dynamic fields [23] A new chaoticparticle swarm optimization algorithm (CPSO) was pro-posed to optimize the control points of the Bezier curve inwhich the total distance between the start point and the endpoint was minimized by using the selected control points[24] Additionally a Bezier-curve-based path planner wasproposed and applied in autonomous vehicles [25] More-over the collaborative collision avoidance method waspresented for multi-incomplete robots based on Bern-steinndashBezier curves [26] Finally an effective and analyticalcontinuous curvature path smoothing algorithm was pro-posed [27] and it was found to be suitable for the sequenceof path points generated by an obstacle avoidance pathplanner

Nonetheless there is still much difficulty in planning asmooth and safe path for a mobile robot by using themethods mentioned above us this paper aims to proposea new smooth path planning method to deal with the re-dundant nodes and peak inflection points in the pathplanning process First genetic operators are used to obtainthe control points of the Bezier curve which ensures thesmooth continuity of the path Second the length of theBezier curve is selected to be the optimization criterion toensure the shortest length of the planned path ird byincreasing the safety distance and the adaptive penalty factorin the fitness function the safety of the robot walking processis guaranteed Finally experimental results illustrate that theproposedmethod can produce a shorter smoother and safereffective path than the existing methods (eg [28ndash31]) Inconclusion the contributions of this paper are as follows (1)compared with the methods introduced in [29 30] ourgenetic operations are used to obtain the control points ofthe Bezier curve which can guarantee the continuity of pathcurvature (2) a shorter smooth path is selected by using anoptimization criterion which can ensure that the generatedpath is optimal without requiring further smoothing and (3)in contrast with the methods in [6 22] the safety in robotsrsquowalking progress is further improved by introducing anadaptive adjustment in the fitness function

e rest of the article is organized as follows e Beziercurve is introduced in Section 2 Section 3 presents the keypoints of the smooth path planning method based on theBezier curve e experimental results on the feasibility andeffectiveness of the proposed method are discussed inSection 4 Finally a conclusion and directions for futurework are presented in Section 5

2 Bezier Curve

e Bezier curve is a continuous smooth curve that is de-termined by a few feature controls points a start point and

an end point e high-order derivative continuity of theBezier curve ensures the smooth variation of the curve fromthe start point to the end point [17 32ndash34] erefore whenthe Bezier curve is used to obtain the traveling path of therobot the planned path is continuous and smooth

Given a curve with m control points P0 P1 P2 Pmthe corresponding Bezier curve is determined as in [35]

P(t) 1113944m

i0B

mi (t)Pi 0le tle 1 (1)

Bmi (t)

m

i1113888 1113889t

i(1 minus t)

mminus i i 0 1 m (2)

where t is the positional parameter (eg when t 025 P(t)

is a quarter of the path from point P0 to Pm) Pi representsthe control point of the Bezier curve and Bm

i (t) is theBernstein polynomial which is the basic function of Beziercurve expressions

According to equations (1) and (2) the derivatives of theBezier curve can be also determined by the control pointse first derivative of a Bezier curve is expressed as follows

_P(t) dP(t)

dt m 1113944

mminus 1

i0B

mminus 1i (t) Pi+1 minus Pi( 1113857 (3)

and the higher-order derivatives of the Bezier curve can becalculated by equation (3) e second derivative of theBezier curve can be expressed as in [35]

euroP(t) m(m minus 1) 1113944mminus 2

i0B

mminus 2i (t) _Pi+2 minus 2 _Pi+1 + _Pi1113872 1113873 (4)

erefore in the two-dimensional plane the curvatureof the Bezier curve with respect to t can be represented as

k(t) 1

R(t)

_Px(t) euroPy(t) minus _Py(t) euroPx(t)

_P2x(t) + _P

2y(t)1113874 1113875

32 (5)

where R(t) stands for the radius of curvature and _Px(t)_Py(t) euroPx(t) and euroPy(t) are the components of the first andsecond derivatives of the Bezier curve P(t) for the X and Y

coordinatesIn general the traditional algorithm of path planning

will produce a lot of sharp inflection points and redundantnodes as shown in Figure 1(a) where S is the start point T isthe end point and P1 P2 P3 P4 P5 are the path inflectionpoints Assuming that P1 P2 P3 P4 P5 are the controlpoints the corresponding Bezier curve can be obtained fromequations (1) and (2) (see Figure 1(b)) From Figure 1(c) itcan be seen that the path obtained by the Bezier curve issmoother and shorter

3 Smooth Path Planning Based on theBezier Curve

In this study the genetic operator and Bezier curve arecombined to find a smooth and safe path for the mobilerobot First the genetic operator is applied to search for the

2 Computational Intelligence and Neuroscience

control points of the Bezier curve then the safety distanceand the adaptive adjustment factor are added to the fitnessfunction to evaluate the safety of the planned Bezier pathLast the shortest path is determined according to the lengthof all the planned Bezier paths e key points of the pro-posed methods are described below

31 Problem Description e path planning problem isgenerally described as searching for a collision-free path fromthe start point to the target point according to certain evalu-ation indicators Specifically the whole planned path from thestart point to the end point must be the shortest and collision-free e problem can be mathematically defined as follows

distance |P(t)|min 0le tle 1 (6)

P(t) isin S⟶ T (7)

P(t) isin collision minus free (8)

where t is the positional parameter |P(t)|min stands for thelength of the Bezier curve path S⟶ T represents the pathfrom start point S to end point T and collision minus free is thepath without collision

Equations (1) and (2) show that the Bezier curve isdetermined by the control points erefore the problem isequal to finding the control points of the Bezier curve withconstraint (9)

Pi notin obs (9)

Here obs is the obstacle and Pi represents the controlpoints of the Bezier curve It must hold that the controlpoints cannot be in the obstacle when searching for thecontrol points of the Bezier curve

32 Chromosome Encoding e chromosome needs to beencoded before genetic manipulation (ie the chromosomeis the control point sequence of the Bezier curve) Commonencoding methods include binary encoding and decimal

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

S

T

P1

P2

P3

P4

P5

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

S

T

(b)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

S

T

P1

P2

P3

P4

P5

(c)

Figure 1 Path planning (a) Traditional path planning (b) Bezier curve path planning (c) comparison of path planning methods

Computational Intelligence and Neuroscience 3

encoding Binary coding uses strings of 0 or 1 to form achromosome which has the advantages of simple operationand easy decoding erefore the chromosome of this paperis encoded in binary

All control points are defined at the center of the grid inthe workspace e transformation from grid numbers tocoordinate values is expressed as

Px(t) mod(num M) minus 05

M minus 05 ifPx(t) minus 051113896 (10)

Py(t) M + ceil(num M) + 05 (11)

where num indicates the grid numberM denotes the size ofmap mod stands for the remainder operation ceil repre-sents rounding down and Px(t) and Py(t) are the X and Ycoordinate components of the center of the gridrespectively

33 Adaptive Adjustment of Fitness Function Although theshortest path can be guaranteed when the path length isregarded as the main criterion there may exist someproblems of collision caused by the small distance betweenthe robot and obstacles erefore this chapter proposes asafety-assurance-based fitness function to increase the safetydistance by introducing an adaptive penalty factor which isexpressed as follows

fitnew

fit1 for feasible paths

1wlowast1 fit1 + wlowast2 fit2

for infeasible paths

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(12)

fit1 |P(t)| (13)

fit2

0 Lmin gtLe

1 minusLminLe

1113874 11138752 Lmin ltLe

⎧⎪⎪⎨

⎪⎪⎩(14)

where P(t) is the length of the Bezier curve path obtained bythe control points sequence under constraints (6)ndash(9) Le isthe safety distance from the obstacle When the minimumdistance Lmin between the path and the obstacle is less thanthe safety distance the penalty will be imposed Equation(14) shows that the intensity of punishment is related to Lmine closer the distance between the path and the obstaclethe higher the penalty strength is enables dynamic ad-justment of the fitness function to improve the quality of theplanning path According to equation (12) we can observethat the shorter the planning path and the higher the fitnessvalue of the path the greater the probability that the path willbe selected

34 Genetic Operation In this study genetic operations areintroduced into the Bezier curve to find the control pointsere are three genetic operators the selection operatorcrossover operator and mutation operator

e selection operator implements the selection of thepath by utilizing different selection strategies according tothe fitness value Selection methods are the roulette methodchampionship method etc is paper adopts the roulettemethod to select the path Assuming that the path length ofthe ith Bezier curve is |Pi(t)| the fitness value of the Beziercurve is fiti (see equation (12)) e probability of the se-lected path is

pi fiti

1113936ni1 fiti

(15)

where n is the number of Bezier curvese crossover operation combines the characteristics of

two parent chromosomes to produce two offspringAccording to the crossover probability the genes of twochromosomes are swapped at a randomly generatedcrossover point A single-point crossover is adopted in thispaper as is shown in Figure 2

e mutation operator is introduced to increase thediversity of solutions It changes the part of the gene bymutating any gene excluding the start point and the endpoint in the chromosome and it avoids premature con-vergence caused by the algorithm falling into local optimumin the solution process In this paper the probability of themutation operator is set to 01 which can guarantee thestability of the algorithmrsquos solution process

4 Experiment and Analysis

In this section to verify the feasibility and effectiveness of theproposed algorithm the path planning of the mobile robot isvalidated in two different grid environments as shown inFigure 3 In environment 1 the starting coordinate of therobot is (00) the end point coordinate is (20 20) theobstacle coverage rate is 1500 and the parameters are setas follows the population size is taken as Q 100 themaximum generation is taken as gn 50 the crossoverprobability is taken as Re 09 and themutation probabilityis taken as Mu 01 w1 08 w2 01 and Le 025 Inenvironment 2 the starting coordinate of the robot is (00)the end point coordinate is (2020) the obstacle coveragerate is 2075 and the parameters are set as follows thepopulation size is taken as Q 100 the maximum gener-ation is taken as gn 100 the crossover probability is takenas Re 09 and the mutation probability is taken asMu 01 w1 09 w2 01 and Le 025

41 Feasibility Experiment of Bezier Smoothing AlgorithmsIn the experiments the traditional ant colony optimization(ACO) the traditional genetic algorithm (GA) ant colonyoptimization combined with the genetic algorithm (ACO-GA) the artificial fish swarm algorithm (ASFA-GA) theBezier curve smoothing algorithm (BCA) and the Beziersmoothing algorithm with increased safety distance (BCA-Q) are adopted to conduct experiments in environment 1Figure 4 shows the path planning results with different al-gorithms In order to eliminate the influence of random andother contingency factors on the algorithm the above al-gorithms are executed 30 times independently and the

4 Computational Intelligence and Neuroscience

statistical results are given in Table 1 where ldquomdashrdquo means thatstatistical results cannot be obtained

Combining Figure 4 and Table 1 the following can befound in terms of path length (1) ACO the GA and theASFA-GA can plan a collision-free path (as shown inFigures 4(a) 4(b) and 4(d)) with the path lengths being346248 328663 and 312144 respectively Compared withthe BCA (299416) the planning paths of ACO the GA andthe ASFA-GA are longer which is caused by a lot of re-dundant nodes and redundant infection points in the path(2) When the GA is introduced into ACO ie for the ACO-GA the path length is 317964 this is an improved per-formance compared with just ACO and just the GAHowever there are still peak inflection points (egFigure 4(c)) (3) It can be seen from Figures 4(e) and 4(f)that the quality of the path obtained by the BCA and BCA-Q

has been significantly improved here the blue circles are thecontrol points of the Bezier curve and the red line is theoptimal smooth path e path lengths are 299416 and311843 respectively Combined with Table 1 the minimumand average values of the planned path lengths obtained byusing the BCA and BCA-Q are better than those of othermethods e reason for this is that the improved algorithmreduces redundant inflection points and redundant nodes inthe path planning therefore the path is smoother andshorter (4) Although the path length planned in Figure 4(f )is longer than that shown in Figure 4(e) the mobile securityof the robot is guaranteed and the path planning perfor-mance is better than that of other algorithms

In terms of running time Table 1 shows that theASFA-GA is the best algorithm in terms of required timefor simulations Moreover the ACO-GA has increased

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

Intersection (7 8)

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

Intersection (7 8)

(b)

Figure 2 Single-point crossover operation where intersection (7 8) is a crossover point (a) Control points before crossing (b) controlpoints after crossing

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(a)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

Figure 3 Path planning environments (a) environment 1 (b) environment 2

Computational Intelligence and Neuroscience 5

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(a)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(c)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(d)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(e)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(f )

Figure 4 Path planning simulation results in environment 1 (a) Path planned by ACO (b) path planned by the GA (c) path planned by theACO-GA (d) path planned by the ASFA-GA (e) path planned by the BCA (f ) path planned by the BCA-Q

6 Computational Intelligence and Neuroscience

Table 1 Path planning statistics in environment 1

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 346248 503794 425416 104718 204864 156471GA 328663 386874 357144 504718 701453 614772ACO-GA 317964 354147 334172 595674 844792 673358ASFA-GA 312144 335479 323648 84716 mdash mdashBCA 299416 324471 309436 1364794 2641479 2024861BCA-Q 311843 348517 329973 1746659 2963371 2344781

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

20

(a)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

20

(c)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(d)

Figure 5 Continued

Computational Intelligence and Neuroscience 7

simulation time due to the introduction of the GA is isbecause the GA introduces crossover and mutation op-erations in the process of path planning Finally Table 1shows that although the simulation times of the BCA andBCA-Q are longer than those of other algorithms theirminimum path lengths are shorter erefore comparedwith the traditional algorithms the proposed algorithmhas better performance in terms of path length andsmoothness in the environment with low real-timerequirements

42 Effectiveness Experiment of Bezier Smoothing AlgorithmsIn order to verify the performance of the proposed algo-rithm a complex grid environment of 20 times 20 is establishedin this paper e algorithms shown in Section 41 are ap-plied in environment 2 Figure 5 shows the path planningresults with different algorithms In order to eliminate theinfluence of random and other contingency factors on thealgorithm the aforementioned algorithms are executed 30times independently and the statistical results are recordedin Table 2 where ldquomdashrdquomeans that statistical results cannot beobtained

Based on the analysis of Figure 5 and Table 2 it can beobserved that the obstacle coverage of environment 2 is

increased by 575 compared with environment 1However the added coverage makes it more difficult tosearch for the global optimal solution Compared withTable 1 the simulation time and distance of path planningin Table 2 are increasede path length of the ASFA-GA is323821 Although the path is not the longest of all al-gorithms there are obvious inflection points (seeFigure 5(d)) (2) From Figures 5(a)ndash5(d) and Table 2 onecan see that ACO and the ACO-GA can also find a col-lision-free path in complex environments but their av-erage path length is longer compared to the ASFA-GAwhich is 336631 We can also observe from Figure 5(e)that the path obtained by the BCA is smoother and shortercompared with those obtained by other algorithms Re-dundant nodes are avoided and the path length is 303458From Table 2 we can see that the BCA is slightly insuf-ficient compared to other algorithms in terms of simu-lation time consumption However it is still effective incomplex environments from the perspective of path lengthand smoothness Additionally although the inflectionpoint disappears in the path planned by the BCA the pathis very close to the obstacle By increasing the safetydistance in the fitness function the BCA-Q improves thesecurity of the mobile robot which can be verified inFigure 5(f ) However this advantage is achieved by

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(e)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(f )

Figure 5 Path planning simulation results in environment 2 (a) Path planned by ACO (b) path planned by the GA (c) path planned by theACO-GA (d) path planned by the ASFA-GA (e) path planned by the BCA (f ) path planned by the BCA-Q

Table 2 Path planning statistics in environment 2

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 354526 561147 459981 257242 387413 325469GA 337963 404483 375448 857157 1076814 961479ACO-GA 329688 373691 351472 1084743 1364731 1211173ASFA-GA 323821 347728 336631 204473 mdash mdashBCA 303458 337694 321473 2027781 3789283 2934417BCA-Q 318503 354473 337628 2865575 4026491 3454799

8 Computational Intelligence and Neuroscience

sacrificing the path length e planned path length of theBCA-Q is 318503 which is 496 greater than that of theBCA It is noteworthy that the path length of BCA-Q is stillshorter than those of ACO and the GA Moreover itssmoothness and path security are greatly improved

5 Conclusions

In this paper a new smooth path planning method com-bining genetic operators and Bezier curves has been pro-posed for mobile robots First the control points of theBezier curve were determined by a genetic operation and thesmoothing characteristics of the Bezier curve were used tomake the planning path smoother and more consistentwhich reduced the energy loss in robot movement Secondthe safety distance was added to the fitness function andcould be dynamically adjusted according to the distancebetween the path and the obstacle to ensure the safe andefficient movement of the robot Finally the simulationresults showed that the proposed algorithm was effective infinding an optimal path this optimal path was shortersmoother and safer than those obtained by traditional al-gorithms However the simulation time was increased iswas because the improved algorithm paid more attention tothe quality of the solution in the process of path planningere still exist various meaningful topics to be addressedsuch as how to select the most appropriate number ofcontrol points of the continuous Bezier curve how to applythe algorithm to more complex practical environments(such as rooms or other special environments) and how toimprove the calculation efficiency with more control points

Data Availability

e data used to support the findings of this study are in-cluded within the article All required models and param-eters are listed in the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported in part by the National Key Re-search and Development Program of China under Grant2016YFE0104600 and the Science and Technology Programof Henan Province of China under Grant 172102410071

References

[1] M A P Garcia O Montiel O Castillo R Sepulveda andPMelin ldquoPath planning for autonomousmobile robot navigationwith ant colony optimization and fuzzy cost function evaluationrdquoApplied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[2] G Li andW Chou ldquoPath planning for mobile robot using self-adaptive learning particle swarm optimizationrdquo Science ChinaInformation Sciences vol 61 no 5 Article ID 052204 2018

[3] R K Mandava S Bondada and P R Vundavilli ldquoAn op-timized path planning for the mobile robot using potential

field method and PSO algorithmrdquo Advances in IntelligentSystems and Computing vol 8 no 7 pp 139ndash150 2019

[4] R Li W Wu and H Qiao ldquoe compliance of robotichandsmdashfrom functionality to mechanismrdquo Assembly Auto-mation vol 35 no 3 pp 281ndash286 2015

[5] D C Robinson D A Sanders and EMazharsolook ldquoAmbientintelligence for optimal manufacturing and energy efficiencyrdquoAssembly Automation vol 35 no 3 pp 234ndash248 2015

[6] B Song Z Wang and L Sheng ldquoA new genetic algorithmapproach to smooth path planning for mobile robotsrdquo As-sembly Automation vol 36 no 2 pp 138ndash145 2016

[7] A Bakdi A Hentout H Boutami A Maoudj O Hachourand B Bouzouia ldquoOptimal path planning and execution formobile robots using genetic algorithm and adaptive fuzzy-logic controlrdquo Robotics and Autonomous Systems vol 89pp 95ndash109 2017

[8] Y Deng Y Liu and D Zhou ldquoAn improved genetic algo-rithm with initial population strategy for symmetric TSPrdquoMathematical Problems in Engineering vol 2015 Article ID212794 6 pages 2015

[9] B K Patle D R K Parhi A Jagadeesh and S K KashyapldquoMatrix-binary codes based genetic algorithm for pathplanning of mobile robotrdquo Computers amp Electrical Engi-neering vol 67 pp 708ndash728 2018

[10] M A Contreras-Cruz V Ayala-Ramirez and U H Hernandez-Belmonte ldquoMobile robot path planning using artificial beecolony and evolutionary programmingrdquo Applied Soft Comput-ing vol 30 pp 319ndash328 2015

[11] D Whitley D Hains and A Howe ldquoA hybrid genetic al-gorithm for the traveling salesman problem using generalizedpartition crossoverrdquo in Proceedings of the InternationalConference on Parallel Problem Solving from Naturepp 566ndash575 Springer Krakow Poland September 2010

[12] M Albayrak and N Allahverdi ldquoDevelopment a new mu-tation operator to solve the traveling salesman problem by aidof genetic algorithmsrdquo Expert Systems with Applicationsvol 38 no 3 pp 1313ndash1320 2011

[13] G Dantzig R Fulkerson and S Johnson ldquoSolution of a large-scale traveling-salesman problemrdquo Journal of the OperationsResearch Society of America vol 2 no 4 pp 393ndash410 1954

[14] M Nazarahari E Khanmirza and S Doostie ldquoMulti-ob-jective multi-robot path planning in continuous environmentusing an enhanced genetic algorithmrdquo Expert Systems withApplications vol 115 pp 106ndash120 2019

[15] J Ni K Wang H Huang et al ldquoRobot path planning basedon an improved genetic algorithm with variable lengthchromosomerdquo in Proceedings of the 2016 12th InternationalConference on Natural Computation Fuzzy Systems andKnowledge Discovery (ICNC-FSKD) pp 145ndash149 ChangshaChina August 2016

[16] C-C Tsai H-C Huang and C-K Chan ldquoParallel elite ge-netic algorithm and its application to global path planning forautonomous robot navigationrdquo IEEE Transactions on In-dustrial Electronics vol 58 no 10 pp 4813ndash4821 2011

[17] M M Malakiyeh S Shojaee and S Hamzehei JavaranldquoInsight into an implicit time integration method based onBezier curve and third-order Bernstein basis function forstructural dynamicsrdquo Asian Journal of Civil Engineeringvol 19 no 1 pp 1ndash11 2018

[18] N Arana-Daniel A A Gallegos C Lopez-Franco et alldquoSmooth global and local path planning for mobile robotusing particle swarm optimization radial basis functionssplines and Bezier curvesrdquo in Proceedings of the 2014 IEEE

Computational Intelligence and Neuroscience 9

Congress on Evolutionary Computation (CEC) pp 175ndash182Beijing China July 2014

[19] A A Neto DGMacharet andM FM Campos ldquoFeasible RRT-based path planning using seventh order Bezier curvesrdquo in Pro-ceedings of the IEEE International Conference on Intelligent Robotsand Systems pp 1445ndash1450 Taipei Taiwan October 2010

[20] B Song G Tian and F Zhou ldquoA comparison study on pathsmoothing algorithms for laser robot navigated mobile robotpath planning in intelligent spacerdquo Journal of Information andComputational Science vol 7 no 1 pp 2943ndash2950 2010

[21] K G Jolly R Sreerama Kumar and R Vijayakumar ldquoA Beziercurve based path planning in a multi-agent robot soccer systemwithout violating the acceleration limitsrdquo Robotics and Au-tonomous Systems vol 57 no 1 pp 23ndash33 2009

[22] Y J Ho and J S Liu ldquoCollision-free curvature-boundedsmooth path planning using composite Bezier curve based onVoronoi diagramrdquo in Proceedings of the 2009 IEEE Inter-national Symposium on Computational Intelligence in Roboticsand Automation-(CIRA) pp 463ndash468 Daejeon South KoreaDecember 2009

[23] M Elhoseny A arwat and A E Hassanien ldquoBezier curvebased path planning in a dynamic field using modified geneticalgorithmrdquo Journal of Computational Science vol 25pp 339ndash350 2018

[24] A arwat M Elhoseny A E Hassanien et al ldquoIntelligentBezier curve-based path planningmodel using chaotic particleswarm optimization algorithmrdquo Cluster Computing vol 22no S2 pp 4745ndash4766 2019

[25] L Han H Yashiro H T N Nejad et al ldquoBezier curve basedpath planning for autonomous vehicle in urban environ-mentrdquo in Proceedings of the 2010 IEEE Intelligent VehiclesSymposium pp 1036ndash1042 San Diego CA USA June 2010

[26] I Skrjanc and G Klancar ldquoOptimal cooperative collisionavoidance betweenmultiple robots based on BernsteinndashBeziercurvesrdquo Robotics and Autonomous Systems vol 58 no 1pp 1ndash9 2010

[27] K Yang and S Sukkarieh ldquoAn analytical continuous-cur-vature path-smoothing algorithmrdquo IEEE Transactions onRobotics vol 26 no 3 pp 561ndash568 2010

[28] S H Chia K L Su J H Guo et al ldquoAnt colony system basedmobile robot path planningrdquo in Proceedings of the 2010 FourthInternational Conference on Genetic and Evolutionary Com-puting pp 210ndash213 Shenzhen China December 2010

[29] A T Ismail A Sheta and M Al-Weshah ldquoA mobile robotpath planning using genetic algorithm in static environmentrdquoJournal of Computer Science vol 4 no 4 pp 341ndash344 2008

[30] I Chaari A Koubaa H Bennaceur et al ldquoSmartPATH ahybrid ACO-GA algorithm for robot path planningrdquo inProceedings of the 2012 IEEE Congress on EvolutionaryComputation pp 1ndash8 Brisbane QLD Australia June 2012

[31] K Liang Z J Chen and X Q Yan ldquoMobile robot pathplanning simulation researchrdquoModern Electronic Technologyvol 41 no 17 pp 167ndash172 2018

[32] MMMalakiyeh S Shojaee and S H Javaran ldquoDevelopmentof a direct time integration method based on Bezier curve and5th-order Bernstein basis functionrdquo Computers amp Structuresvol 194 pp 15ndash31 2018

[33] N Wang H Guo B Chen et al ldquoDesign of a rotary dielectricelastomer actuator using a topology optimization methodbased on pairs of curvesrdquo Smart Materials and Structuresvol 27 no 5 Article ID 055011 2018

[34] B Song Z Wang L Zou L Xu and F E Alsaadi ldquoA newapproach to smooth global path planning of mobile robots

with kinematic constraintsrdquo International Journal of MachineLearning and Cybernetics vol 10 no 1 pp 107ndash119 2019

[35] B Song Z Wang L Zou et al ldquoOn global smooth pathplanning for mobile robots using a novel multimodal delayedPSO algorithmrdquo Cognitive Computation vol 9 no 1pp 5ndash17 2017

10 Computational Intelligence and Neuroscience

control points of the Bezier curve then the safety distanceand the adaptive adjustment factor are added to the fitnessfunction to evaluate the safety of the planned Bezier pathLast the shortest path is determined according to the lengthof all the planned Bezier paths e key points of the pro-posed methods are described below

31 Problem Description e path planning problem isgenerally described as searching for a collision-free path fromthe start point to the target point according to certain evalu-ation indicators Specifically the whole planned path from thestart point to the end point must be the shortest and collision-free e problem can be mathematically defined as follows

distance |P(t)|min 0le tle 1 (6)

P(t) isin S⟶ T (7)

P(t) isin collision minus free (8)

where t is the positional parameter |P(t)|min stands for thelength of the Bezier curve path S⟶ T represents the pathfrom start point S to end point T and collision minus free is thepath without collision

Equations (1) and (2) show that the Bezier curve isdetermined by the control points erefore the problem isequal to finding the control points of the Bezier curve withconstraint (9)

Pi notin obs (9)

Here obs is the obstacle and Pi represents the controlpoints of the Bezier curve It must hold that the controlpoints cannot be in the obstacle when searching for thecontrol points of the Bezier curve

32 Chromosome Encoding e chromosome needs to beencoded before genetic manipulation (ie the chromosomeis the control point sequence of the Bezier curve) Commonencoding methods include binary encoding and decimal

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

S

T

P1

P2

P3

P4

P5

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

S

T

(b)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

S

T

P1

P2

P3

P4

P5

(c)

Figure 1 Path planning (a) Traditional path planning (b) Bezier curve path planning (c) comparison of path planning methods

Computational Intelligence and Neuroscience 3

encoding Binary coding uses strings of 0 or 1 to form achromosome which has the advantages of simple operationand easy decoding erefore the chromosome of this paperis encoded in binary

All control points are defined at the center of the grid inthe workspace e transformation from grid numbers tocoordinate values is expressed as

Px(t) mod(num M) minus 05

M minus 05 ifPx(t) minus 051113896 (10)

Py(t) M + ceil(num M) + 05 (11)

where num indicates the grid numberM denotes the size ofmap mod stands for the remainder operation ceil repre-sents rounding down and Px(t) and Py(t) are the X and Ycoordinate components of the center of the gridrespectively

33 Adaptive Adjustment of Fitness Function Although theshortest path can be guaranteed when the path length isregarded as the main criterion there may exist someproblems of collision caused by the small distance betweenthe robot and obstacles erefore this chapter proposes asafety-assurance-based fitness function to increase the safetydistance by introducing an adaptive penalty factor which isexpressed as follows

fitnew

fit1 for feasible paths

1wlowast1 fit1 + wlowast2 fit2

for infeasible paths

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(12)

fit1 |P(t)| (13)

fit2

0 Lmin gtLe

1 minusLminLe

1113874 11138752 Lmin ltLe

⎧⎪⎪⎨

⎪⎪⎩(14)

where P(t) is the length of the Bezier curve path obtained bythe control points sequence under constraints (6)ndash(9) Le isthe safety distance from the obstacle When the minimumdistance Lmin between the path and the obstacle is less thanthe safety distance the penalty will be imposed Equation(14) shows that the intensity of punishment is related to Lmine closer the distance between the path and the obstaclethe higher the penalty strength is enables dynamic ad-justment of the fitness function to improve the quality of theplanning path According to equation (12) we can observethat the shorter the planning path and the higher the fitnessvalue of the path the greater the probability that the path willbe selected

34 Genetic Operation In this study genetic operations areintroduced into the Bezier curve to find the control pointsere are three genetic operators the selection operatorcrossover operator and mutation operator

e selection operator implements the selection of thepath by utilizing different selection strategies according tothe fitness value Selection methods are the roulette methodchampionship method etc is paper adopts the roulettemethod to select the path Assuming that the path length ofthe ith Bezier curve is |Pi(t)| the fitness value of the Beziercurve is fiti (see equation (12)) e probability of the se-lected path is

pi fiti

1113936ni1 fiti

(15)

where n is the number of Bezier curvese crossover operation combines the characteristics of

two parent chromosomes to produce two offspringAccording to the crossover probability the genes of twochromosomes are swapped at a randomly generatedcrossover point A single-point crossover is adopted in thispaper as is shown in Figure 2

e mutation operator is introduced to increase thediversity of solutions It changes the part of the gene bymutating any gene excluding the start point and the endpoint in the chromosome and it avoids premature con-vergence caused by the algorithm falling into local optimumin the solution process In this paper the probability of themutation operator is set to 01 which can guarantee thestability of the algorithmrsquos solution process

4 Experiment and Analysis

In this section to verify the feasibility and effectiveness of theproposed algorithm the path planning of the mobile robot isvalidated in two different grid environments as shown inFigure 3 In environment 1 the starting coordinate of therobot is (00) the end point coordinate is (20 20) theobstacle coverage rate is 1500 and the parameters are setas follows the population size is taken as Q 100 themaximum generation is taken as gn 50 the crossoverprobability is taken as Re 09 and themutation probabilityis taken as Mu 01 w1 08 w2 01 and Le 025 Inenvironment 2 the starting coordinate of the robot is (00)the end point coordinate is (2020) the obstacle coveragerate is 2075 and the parameters are set as follows thepopulation size is taken as Q 100 the maximum gener-ation is taken as gn 100 the crossover probability is takenas Re 09 and the mutation probability is taken asMu 01 w1 09 w2 01 and Le 025

41 Feasibility Experiment of Bezier Smoothing AlgorithmsIn the experiments the traditional ant colony optimization(ACO) the traditional genetic algorithm (GA) ant colonyoptimization combined with the genetic algorithm (ACO-GA) the artificial fish swarm algorithm (ASFA-GA) theBezier curve smoothing algorithm (BCA) and the Beziersmoothing algorithm with increased safety distance (BCA-Q) are adopted to conduct experiments in environment 1Figure 4 shows the path planning results with different al-gorithms In order to eliminate the influence of random andother contingency factors on the algorithm the above al-gorithms are executed 30 times independently and the

4 Computational Intelligence and Neuroscience

statistical results are given in Table 1 where ldquomdashrdquo means thatstatistical results cannot be obtained

Combining Figure 4 and Table 1 the following can befound in terms of path length (1) ACO the GA and theASFA-GA can plan a collision-free path (as shown inFigures 4(a) 4(b) and 4(d)) with the path lengths being346248 328663 and 312144 respectively Compared withthe BCA (299416) the planning paths of ACO the GA andthe ASFA-GA are longer which is caused by a lot of re-dundant nodes and redundant infection points in the path(2) When the GA is introduced into ACO ie for the ACO-GA the path length is 317964 this is an improved per-formance compared with just ACO and just the GAHowever there are still peak inflection points (egFigure 4(c)) (3) It can be seen from Figures 4(e) and 4(f)that the quality of the path obtained by the BCA and BCA-Q

has been significantly improved here the blue circles are thecontrol points of the Bezier curve and the red line is theoptimal smooth path e path lengths are 299416 and311843 respectively Combined with Table 1 the minimumand average values of the planned path lengths obtained byusing the BCA and BCA-Q are better than those of othermethods e reason for this is that the improved algorithmreduces redundant inflection points and redundant nodes inthe path planning therefore the path is smoother andshorter (4) Although the path length planned in Figure 4(f )is longer than that shown in Figure 4(e) the mobile securityof the robot is guaranteed and the path planning perfor-mance is better than that of other algorithms

In terms of running time Table 1 shows that theASFA-GA is the best algorithm in terms of required timefor simulations Moreover the ACO-GA has increased

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

Intersection (7 8)

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

Intersection (7 8)

(b)

Figure 2 Single-point crossover operation where intersection (7 8) is a crossover point (a) Control points before crossing (b) controlpoints after crossing

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(a)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

Figure 3 Path planning environments (a) environment 1 (b) environment 2

Computational Intelligence and Neuroscience 5

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(a)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(c)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(d)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(e)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(f )

Figure 4 Path planning simulation results in environment 1 (a) Path planned by ACO (b) path planned by the GA (c) path planned by theACO-GA (d) path planned by the ASFA-GA (e) path planned by the BCA (f ) path planned by the BCA-Q

6 Computational Intelligence and Neuroscience

Table 1 Path planning statistics in environment 1

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 346248 503794 425416 104718 204864 156471GA 328663 386874 357144 504718 701453 614772ACO-GA 317964 354147 334172 595674 844792 673358ASFA-GA 312144 335479 323648 84716 mdash mdashBCA 299416 324471 309436 1364794 2641479 2024861BCA-Q 311843 348517 329973 1746659 2963371 2344781

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

20

(a)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

20

(c)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(d)

Figure 5 Continued

Computational Intelligence and Neuroscience 7

simulation time due to the introduction of the GA is isbecause the GA introduces crossover and mutation op-erations in the process of path planning Finally Table 1shows that although the simulation times of the BCA andBCA-Q are longer than those of other algorithms theirminimum path lengths are shorter erefore comparedwith the traditional algorithms the proposed algorithmhas better performance in terms of path length andsmoothness in the environment with low real-timerequirements

42 Effectiveness Experiment of Bezier Smoothing AlgorithmsIn order to verify the performance of the proposed algo-rithm a complex grid environment of 20 times 20 is establishedin this paper e algorithms shown in Section 41 are ap-plied in environment 2 Figure 5 shows the path planningresults with different algorithms In order to eliminate theinfluence of random and other contingency factors on thealgorithm the aforementioned algorithms are executed 30times independently and the statistical results are recordedin Table 2 where ldquomdashrdquomeans that statistical results cannot beobtained

Based on the analysis of Figure 5 and Table 2 it can beobserved that the obstacle coverage of environment 2 is

increased by 575 compared with environment 1However the added coverage makes it more difficult tosearch for the global optimal solution Compared withTable 1 the simulation time and distance of path planningin Table 2 are increasede path length of the ASFA-GA is323821 Although the path is not the longest of all al-gorithms there are obvious inflection points (seeFigure 5(d)) (2) From Figures 5(a)ndash5(d) and Table 2 onecan see that ACO and the ACO-GA can also find a col-lision-free path in complex environments but their av-erage path length is longer compared to the ASFA-GAwhich is 336631 We can also observe from Figure 5(e)that the path obtained by the BCA is smoother and shortercompared with those obtained by other algorithms Re-dundant nodes are avoided and the path length is 303458From Table 2 we can see that the BCA is slightly insuf-ficient compared to other algorithms in terms of simu-lation time consumption However it is still effective incomplex environments from the perspective of path lengthand smoothness Additionally although the inflectionpoint disappears in the path planned by the BCA the pathis very close to the obstacle By increasing the safetydistance in the fitness function the BCA-Q improves thesecurity of the mobile robot which can be verified inFigure 5(f ) However this advantage is achieved by

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(e)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(f )

Figure 5 Path planning simulation results in environment 2 (a) Path planned by ACO (b) path planned by the GA (c) path planned by theACO-GA (d) path planned by the ASFA-GA (e) path planned by the BCA (f ) path planned by the BCA-Q

Table 2 Path planning statistics in environment 2

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 354526 561147 459981 257242 387413 325469GA 337963 404483 375448 857157 1076814 961479ACO-GA 329688 373691 351472 1084743 1364731 1211173ASFA-GA 323821 347728 336631 204473 mdash mdashBCA 303458 337694 321473 2027781 3789283 2934417BCA-Q 318503 354473 337628 2865575 4026491 3454799

8 Computational Intelligence and Neuroscience

sacrificing the path length e planned path length of theBCA-Q is 318503 which is 496 greater than that of theBCA It is noteworthy that the path length of BCA-Q is stillshorter than those of ACO and the GA Moreover itssmoothness and path security are greatly improved

5 Conclusions

In this paper a new smooth path planning method com-bining genetic operators and Bezier curves has been pro-posed for mobile robots First the control points of theBezier curve were determined by a genetic operation and thesmoothing characteristics of the Bezier curve were used tomake the planning path smoother and more consistentwhich reduced the energy loss in robot movement Secondthe safety distance was added to the fitness function andcould be dynamically adjusted according to the distancebetween the path and the obstacle to ensure the safe andefficient movement of the robot Finally the simulationresults showed that the proposed algorithm was effective infinding an optimal path this optimal path was shortersmoother and safer than those obtained by traditional al-gorithms However the simulation time was increased iswas because the improved algorithm paid more attention tothe quality of the solution in the process of path planningere still exist various meaningful topics to be addressedsuch as how to select the most appropriate number ofcontrol points of the continuous Bezier curve how to applythe algorithm to more complex practical environments(such as rooms or other special environments) and how toimprove the calculation efficiency with more control points

Data Availability

e data used to support the findings of this study are in-cluded within the article All required models and param-eters are listed in the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported in part by the National Key Re-search and Development Program of China under Grant2016YFE0104600 and the Science and Technology Programof Henan Province of China under Grant 172102410071

References

[1] M A P Garcia O Montiel O Castillo R Sepulveda andPMelin ldquoPath planning for autonomousmobile robot navigationwith ant colony optimization and fuzzy cost function evaluationrdquoApplied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[2] G Li andW Chou ldquoPath planning for mobile robot using self-adaptive learning particle swarm optimizationrdquo Science ChinaInformation Sciences vol 61 no 5 Article ID 052204 2018

[3] R K Mandava S Bondada and P R Vundavilli ldquoAn op-timized path planning for the mobile robot using potential

field method and PSO algorithmrdquo Advances in IntelligentSystems and Computing vol 8 no 7 pp 139ndash150 2019

[4] R Li W Wu and H Qiao ldquoe compliance of robotichandsmdashfrom functionality to mechanismrdquo Assembly Auto-mation vol 35 no 3 pp 281ndash286 2015

[5] D C Robinson D A Sanders and EMazharsolook ldquoAmbientintelligence for optimal manufacturing and energy efficiencyrdquoAssembly Automation vol 35 no 3 pp 234ndash248 2015

[6] B Song Z Wang and L Sheng ldquoA new genetic algorithmapproach to smooth path planning for mobile robotsrdquo As-sembly Automation vol 36 no 2 pp 138ndash145 2016

[7] A Bakdi A Hentout H Boutami A Maoudj O Hachourand B Bouzouia ldquoOptimal path planning and execution formobile robots using genetic algorithm and adaptive fuzzy-logic controlrdquo Robotics and Autonomous Systems vol 89pp 95ndash109 2017

[8] Y Deng Y Liu and D Zhou ldquoAn improved genetic algo-rithm with initial population strategy for symmetric TSPrdquoMathematical Problems in Engineering vol 2015 Article ID212794 6 pages 2015

[9] B K Patle D R K Parhi A Jagadeesh and S K KashyapldquoMatrix-binary codes based genetic algorithm for pathplanning of mobile robotrdquo Computers amp Electrical Engi-neering vol 67 pp 708ndash728 2018

[10] M A Contreras-Cruz V Ayala-Ramirez and U H Hernandez-Belmonte ldquoMobile robot path planning using artificial beecolony and evolutionary programmingrdquo Applied Soft Comput-ing vol 30 pp 319ndash328 2015

[11] D Whitley D Hains and A Howe ldquoA hybrid genetic al-gorithm for the traveling salesman problem using generalizedpartition crossoverrdquo in Proceedings of the InternationalConference on Parallel Problem Solving from Naturepp 566ndash575 Springer Krakow Poland September 2010

[12] M Albayrak and N Allahverdi ldquoDevelopment a new mu-tation operator to solve the traveling salesman problem by aidof genetic algorithmsrdquo Expert Systems with Applicationsvol 38 no 3 pp 1313ndash1320 2011

[13] G Dantzig R Fulkerson and S Johnson ldquoSolution of a large-scale traveling-salesman problemrdquo Journal of the OperationsResearch Society of America vol 2 no 4 pp 393ndash410 1954

[14] M Nazarahari E Khanmirza and S Doostie ldquoMulti-ob-jective multi-robot path planning in continuous environmentusing an enhanced genetic algorithmrdquo Expert Systems withApplications vol 115 pp 106ndash120 2019

[15] J Ni K Wang H Huang et al ldquoRobot path planning basedon an improved genetic algorithm with variable lengthchromosomerdquo in Proceedings of the 2016 12th InternationalConference on Natural Computation Fuzzy Systems andKnowledge Discovery (ICNC-FSKD) pp 145ndash149 ChangshaChina August 2016

[16] C-C Tsai H-C Huang and C-K Chan ldquoParallel elite ge-netic algorithm and its application to global path planning forautonomous robot navigationrdquo IEEE Transactions on In-dustrial Electronics vol 58 no 10 pp 4813ndash4821 2011

[17] M M Malakiyeh S Shojaee and S Hamzehei JavaranldquoInsight into an implicit time integration method based onBezier curve and third-order Bernstein basis function forstructural dynamicsrdquo Asian Journal of Civil Engineeringvol 19 no 1 pp 1ndash11 2018

[18] N Arana-Daniel A A Gallegos C Lopez-Franco et alldquoSmooth global and local path planning for mobile robotusing particle swarm optimization radial basis functionssplines and Bezier curvesrdquo in Proceedings of the 2014 IEEE

Computational Intelligence and Neuroscience 9

Congress on Evolutionary Computation (CEC) pp 175ndash182Beijing China July 2014

[19] A A Neto DGMacharet andM FM Campos ldquoFeasible RRT-based path planning using seventh order Bezier curvesrdquo in Pro-ceedings of the IEEE International Conference on Intelligent Robotsand Systems pp 1445ndash1450 Taipei Taiwan October 2010

[20] B Song G Tian and F Zhou ldquoA comparison study on pathsmoothing algorithms for laser robot navigated mobile robotpath planning in intelligent spacerdquo Journal of Information andComputational Science vol 7 no 1 pp 2943ndash2950 2010

[21] K G Jolly R Sreerama Kumar and R Vijayakumar ldquoA Beziercurve based path planning in a multi-agent robot soccer systemwithout violating the acceleration limitsrdquo Robotics and Au-tonomous Systems vol 57 no 1 pp 23ndash33 2009

[22] Y J Ho and J S Liu ldquoCollision-free curvature-boundedsmooth path planning using composite Bezier curve based onVoronoi diagramrdquo in Proceedings of the 2009 IEEE Inter-national Symposium on Computational Intelligence in Roboticsand Automation-(CIRA) pp 463ndash468 Daejeon South KoreaDecember 2009

[23] M Elhoseny A arwat and A E Hassanien ldquoBezier curvebased path planning in a dynamic field using modified geneticalgorithmrdquo Journal of Computational Science vol 25pp 339ndash350 2018

[24] A arwat M Elhoseny A E Hassanien et al ldquoIntelligentBezier curve-based path planningmodel using chaotic particleswarm optimization algorithmrdquo Cluster Computing vol 22no S2 pp 4745ndash4766 2019

[25] L Han H Yashiro H T N Nejad et al ldquoBezier curve basedpath planning for autonomous vehicle in urban environ-mentrdquo in Proceedings of the 2010 IEEE Intelligent VehiclesSymposium pp 1036ndash1042 San Diego CA USA June 2010

[26] I Skrjanc and G Klancar ldquoOptimal cooperative collisionavoidance betweenmultiple robots based on BernsteinndashBeziercurvesrdquo Robotics and Autonomous Systems vol 58 no 1pp 1ndash9 2010

[27] K Yang and S Sukkarieh ldquoAn analytical continuous-cur-vature path-smoothing algorithmrdquo IEEE Transactions onRobotics vol 26 no 3 pp 561ndash568 2010

[28] S H Chia K L Su J H Guo et al ldquoAnt colony system basedmobile robot path planningrdquo in Proceedings of the 2010 FourthInternational Conference on Genetic and Evolutionary Com-puting pp 210ndash213 Shenzhen China December 2010

[29] A T Ismail A Sheta and M Al-Weshah ldquoA mobile robotpath planning using genetic algorithm in static environmentrdquoJournal of Computer Science vol 4 no 4 pp 341ndash344 2008

[30] I Chaari A Koubaa H Bennaceur et al ldquoSmartPATH ahybrid ACO-GA algorithm for robot path planningrdquo inProceedings of the 2012 IEEE Congress on EvolutionaryComputation pp 1ndash8 Brisbane QLD Australia June 2012

[31] K Liang Z J Chen and X Q Yan ldquoMobile robot pathplanning simulation researchrdquoModern Electronic Technologyvol 41 no 17 pp 167ndash172 2018

[32] MMMalakiyeh S Shojaee and S H Javaran ldquoDevelopmentof a direct time integration method based on Bezier curve and5th-order Bernstein basis functionrdquo Computers amp Structuresvol 194 pp 15ndash31 2018

[33] N Wang H Guo B Chen et al ldquoDesign of a rotary dielectricelastomer actuator using a topology optimization methodbased on pairs of curvesrdquo Smart Materials and Structuresvol 27 no 5 Article ID 055011 2018

[34] B Song Z Wang L Zou L Xu and F E Alsaadi ldquoA newapproach to smooth global path planning of mobile robots

with kinematic constraintsrdquo International Journal of MachineLearning and Cybernetics vol 10 no 1 pp 107ndash119 2019

[35] B Song Z Wang L Zou et al ldquoOn global smooth pathplanning for mobile robots using a novel multimodal delayedPSO algorithmrdquo Cognitive Computation vol 9 no 1pp 5ndash17 2017

10 Computational Intelligence and Neuroscience

encoding Binary coding uses strings of 0 or 1 to form achromosome which has the advantages of simple operationand easy decoding erefore the chromosome of this paperis encoded in binary

All control points are defined at the center of the grid inthe workspace e transformation from grid numbers tocoordinate values is expressed as

Px(t) mod(num M) minus 05

M minus 05 ifPx(t) minus 051113896 (10)

Py(t) M + ceil(num M) + 05 (11)

where num indicates the grid numberM denotes the size ofmap mod stands for the remainder operation ceil repre-sents rounding down and Px(t) and Py(t) are the X and Ycoordinate components of the center of the gridrespectively

33 Adaptive Adjustment of Fitness Function Although theshortest path can be guaranteed when the path length isregarded as the main criterion there may exist someproblems of collision caused by the small distance betweenthe robot and obstacles erefore this chapter proposes asafety-assurance-based fitness function to increase the safetydistance by introducing an adaptive penalty factor which isexpressed as follows

fitnew

fit1 for feasible paths

1wlowast1 fit1 + wlowast2 fit2

for infeasible paths

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(12)

fit1 |P(t)| (13)

fit2

0 Lmin gtLe

1 minusLminLe

1113874 11138752 Lmin ltLe

⎧⎪⎪⎨

⎪⎪⎩(14)

where P(t) is the length of the Bezier curve path obtained bythe control points sequence under constraints (6)ndash(9) Le isthe safety distance from the obstacle When the minimumdistance Lmin between the path and the obstacle is less thanthe safety distance the penalty will be imposed Equation(14) shows that the intensity of punishment is related to Lmine closer the distance between the path and the obstaclethe higher the penalty strength is enables dynamic ad-justment of the fitness function to improve the quality of theplanning path According to equation (12) we can observethat the shorter the planning path and the higher the fitnessvalue of the path the greater the probability that the path willbe selected

34 Genetic Operation In this study genetic operations areintroduced into the Bezier curve to find the control pointsere are three genetic operators the selection operatorcrossover operator and mutation operator

e selection operator implements the selection of thepath by utilizing different selection strategies according tothe fitness value Selection methods are the roulette methodchampionship method etc is paper adopts the roulettemethod to select the path Assuming that the path length ofthe ith Bezier curve is |Pi(t)| the fitness value of the Beziercurve is fiti (see equation (12)) e probability of the se-lected path is

pi fiti

1113936ni1 fiti

(15)

where n is the number of Bezier curvese crossover operation combines the characteristics of

two parent chromosomes to produce two offspringAccording to the crossover probability the genes of twochromosomes are swapped at a randomly generatedcrossover point A single-point crossover is adopted in thispaper as is shown in Figure 2

e mutation operator is introduced to increase thediversity of solutions It changes the part of the gene bymutating any gene excluding the start point and the endpoint in the chromosome and it avoids premature con-vergence caused by the algorithm falling into local optimumin the solution process In this paper the probability of themutation operator is set to 01 which can guarantee thestability of the algorithmrsquos solution process

4 Experiment and Analysis

In this section to verify the feasibility and effectiveness of theproposed algorithm the path planning of the mobile robot isvalidated in two different grid environments as shown inFigure 3 In environment 1 the starting coordinate of therobot is (00) the end point coordinate is (20 20) theobstacle coverage rate is 1500 and the parameters are setas follows the population size is taken as Q 100 themaximum generation is taken as gn 50 the crossoverprobability is taken as Re 09 and themutation probabilityis taken as Mu 01 w1 08 w2 01 and Le 025 Inenvironment 2 the starting coordinate of the robot is (00)the end point coordinate is (2020) the obstacle coveragerate is 2075 and the parameters are set as follows thepopulation size is taken as Q 100 the maximum gener-ation is taken as gn 100 the crossover probability is takenas Re 09 and the mutation probability is taken asMu 01 w1 09 w2 01 and Le 025

41 Feasibility Experiment of Bezier Smoothing AlgorithmsIn the experiments the traditional ant colony optimization(ACO) the traditional genetic algorithm (GA) ant colonyoptimization combined with the genetic algorithm (ACO-GA) the artificial fish swarm algorithm (ASFA-GA) theBezier curve smoothing algorithm (BCA) and the Beziersmoothing algorithm with increased safety distance (BCA-Q) are adopted to conduct experiments in environment 1Figure 4 shows the path planning results with different al-gorithms In order to eliminate the influence of random andother contingency factors on the algorithm the above al-gorithms are executed 30 times independently and the

4 Computational Intelligence and Neuroscience

statistical results are given in Table 1 where ldquomdashrdquo means thatstatistical results cannot be obtained

Combining Figure 4 and Table 1 the following can befound in terms of path length (1) ACO the GA and theASFA-GA can plan a collision-free path (as shown inFigures 4(a) 4(b) and 4(d)) with the path lengths being346248 328663 and 312144 respectively Compared withthe BCA (299416) the planning paths of ACO the GA andthe ASFA-GA are longer which is caused by a lot of re-dundant nodes and redundant infection points in the path(2) When the GA is introduced into ACO ie for the ACO-GA the path length is 317964 this is an improved per-formance compared with just ACO and just the GAHowever there are still peak inflection points (egFigure 4(c)) (3) It can be seen from Figures 4(e) and 4(f)that the quality of the path obtained by the BCA and BCA-Q

has been significantly improved here the blue circles are thecontrol points of the Bezier curve and the red line is theoptimal smooth path e path lengths are 299416 and311843 respectively Combined with Table 1 the minimumand average values of the planned path lengths obtained byusing the BCA and BCA-Q are better than those of othermethods e reason for this is that the improved algorithmreduces redundant inflection points and redundant nodes inthe path planning therefore the path is smoother andshorter (4) Although the path length planned in Figure 4(f )is longer than that shown in Figure 4(e) the mobile securityof the robot is guaranteed and the path planning perfor-mance is better than that of other algorithms

In terms of running time Table 1 shows that theASFA-GA is the best algorithm in terms of required timefor simulations Moreover the ACO-GA has increased

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

Intersection (7 8)

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

Intersection (7 8)

(b)

Figure 2 Single-point crossover operation where intersection (7 8) is a crossover point (a) Control points before crossing (b) controlpoints after crossing

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(a)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

Figure 3 Path planning environments (a) environment 1 (b) environment 2

Computational Intelligence and Neuroscience 5

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(a)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(c)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(d)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(e)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(f )

Figure 4 Path planning simulation results in environment 1 (a) Path planned by ACO (b) path planned by the GA (c) path planned by theACO-GA (d) path planned by the ASFA-GA (e) path planned by the BCA (f ) path planned by the BCA-Q

6 Computational Intelligence and Neuroscience

Table 1 Path planning statistics in environment 1

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 346248 503794 425416 104718 204864 156471GA 328663 386874 357144 504718 701453 614772ACO-GA 317964 354147 334172 595674 844792 673358ASFA-GA 312144 335479 323648 84716 mdash mdashBCA 299416 324471 309436 1364794 2641479 2024861BCA-Q 311843 348517 329973 1746659 2963371 2344781

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

20

(a)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

20

(c)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(d)

Figure 5 Continued

Computational Intelligence and Neuroscience 7

simulation time due to the introduction of the GA is isbecause the GA introduces crossover and mutation op-erations in the process of path planning Finally Table 1shows that although the simulation times of the BCA andBCA-Q are longer than those of other algorithms theirminimum path lengths are shorter erefore comparedwith the traditional algorithms the proposed algorithmhas better performance in terms of path length andsmoothness in the environment with low real-timerequirements

42 Effectiveness Experiment of Bezier Smoothing AlgorithmsIn order to verify the performance of the proposed algo-rithm a complex grid environment of 20 times 20 is establishedin this paper e algorithms shown in Section 41 are ap-plied in environment 2 Figure 5 shows the path planningresults with different algorithms In order to eliminate theinfluence of random and other contingency factors on thealgorithm the aforementioned algorithms are executed 30times independently and the statistical results are recordedin Table 2 where ldquomdashrdquomeans that statistical results cannot beobtained

Based on the analysis of Figure 5 and Table 2 it can beobserved that the obstacle coverage of environment 2 is

increased by 575 compared with environment 1However the added coverage makes it more difficult tosearch for the global optimal solution Compared withTable 1 the simulation time and distance of path planningin Table 2 are increasede path length of the ASFA-GA is323821 Although the path is not the longest of all al-gorithms there are obvious inflection points (seeFigure 5(d)) (2) From Figures 5(a)ndash5(d) and Table 2 onecan see that ACO and the ACO-GA can also find a col-lision-free path in complex environments but their av-erage path length is longer compared to the ASFA-GAwhich is 336631 We can also observe from Figure 5(e)that the path obtained by the BCA is smoother and shortercompared with those obtained by other algorithms Re-dundant nodes are avoided and the path length is 303458From Table 2 we can see that the BCA is slightly insuf-ficient compared to other algorithms in terms of simu-lation time consumption However it is still effective incomplex environments from the perspective of path lengthand smoothness Additionally although the inflectionpoint disappears in the path planned by the BCA the pathis very close to the obstacle By increasing the safetydistance in the fitness function the BCA-Q improves thesecurity of the mobile robot which can be verified inFigure 5(f ) However this advantage is achieved by

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(e)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(f )

Figure 5 Path planning simulation results in environment 2 (a) Path planned by ACO (b) path planned by the GA (c) path planned by theACO-GA (d) path planned by the ASFA-GA (e) path planned by the BCA (f ) path planned by the BCA-Q

Table 2 Path planning statistics in environment 2

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 354526 561147 459981 257242 387413 325469GA 337963 404483 375448 857157 1076814 961479ACO-GA 329688 373691 351472 1084743 1364731 1211173ASFA-GA 323821 347728 336631 204473 mdash mdashBCA 303458 337694 321473 2027781 3789283 2934417BCA-Q 318503 354473 337628 2865575 4026491 3454799

8 Computational Intelligence and Neuroscience

sacrificing the path length e planned path length of theBCA-Q is 318503 which is 496 greater than that of theBCA It is noteworthy that the path length of BCA-Q is stillshorter than those of ACO and the GA Moreover itssmoothness and path security are greatly improved

5 Conclusions

In this paper a new smooth path planning method com-bining genetic operators and Bezier curves has been pro-posed for mobile robots First the control points of theBezier curve were determined by a genetic operation and thesmoothing characteristics of the Bezier curve were used tomake the planning path smoother and more consistentwhich reduced the energy loss in robot movement Secondthe safety distance was added to the fitness function andcould be dynamically adjusted according to the distancebetween the path and the obstacle to ensure the safe andefficient movement of the robot Finally the simulationresults showed that the proposed algorithm was effective infinding an optimal path this optimal path was shortersmoother and safer than those obtained by traditional al-gorithms However the simulation time was increased iswas because the improved algorithm paid more attention tothe quality of the solution in the process of path planningere still exist various meaningful topics to be addressedsuch as how to select the most appropriate number ofcontrol points of the continuous Bezier curve how to applythe algorithm to more complex practical environments(such as rooms or other special environments) and how toimprove the calculation efficiency with more control points

Data Availability

e data used to support the findings of this study are in-cluded within the article All required models and param-eters are listed in the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported in part by the National Key Re-search and Development Program of China under Grant2016YFE0104600 and the Science and Technology Programof Henan Province of China under Grant 172102410071

References

[1] M A P Garcia O Montiel O Castillo R Sepulveda andPMelin ldquoPath planning for autonomousmobile robot navigationwith ant colony optimization and fuzzy cost function evaluationrdquoApplied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[2] G Li andW Chou ldquoPath planning for mobile robot using self-adaptive learning particle swarm optimizationrdquo Science ChinaInformation Sciences vol 61 no 5 Article ID 052204 2018

[3] R K Mandava S Bondada and P R Vundavilli ldquoAn op-timized path planning for the mobile robot using potential

field method and PSO algorithmrdquo Advances in IntelligentSystems and Computing vol 8 no 7 pp 139ndash150 2019

[4] R Li W Wu and H Qiao ldquoe compliance of robotichandsmdashfrom functionality to mechanismrdquo Assembly Auto-mation vol 35 no 3 pp 281ndash286 2015

[5] D C Robinson D A Sanders and EMazharsolook ldquoAmbientintelligence for optimal manufacturing and energy efficiencyrdquoAssembly Automation vol 35 no 3 pp 234ndash248 2015

[6] B Song Z Wang and L Sheng ldquoA new genetic algorithmapproach to smooth path planning for mobile robotsrdquo As-sembly Automation vol 36 no 2 pp 138ndash145 2016

[7] A Bakdi A Hentout H Boutami A Maoudj O Hachourand B Bouzouia ldquoOptimal path planning and execution formobile robots using genetic algorithm and adaptive fuzzy-logic controlrdquo Robotics and Autonomous Systems vol 89pp 95ndash109 2017

[8] Y Deng Y Liu and D Zhou ldquoAn improved genetic algo-rithm with initial population strategy for symmetric TSPrdquoMathematical Problems in Engineering vol 2015 Article ID212794 6 pages 2015

[9] B K Patle D R K Parhi A Jagadeesh and S K KashyapldquoMatrix-binary codes based genetic algorithm for pathplanning of mobile robotrdquo Computers amp Electrical Engi-neering vol 67 pp 708ndash728 2018

[10] M A Contreras-Cruz V Ayala-Ramirez and U H Hernandez-Belmonte ldquoMobile robot path planning using artificial beecolony and evolutionary programmingrdquo Applied Soft Comput-ing vol 30 pp 319ndash328 2015

[11] D Whitley D Hains and A Howe ldquoA hybrid genetic al-gorithm for the traveling salesman problem using generalizedpartition crossoverrdquo in Proceedings of the InternationalConference on Parallel Problem Solving from Naturepp 566ndash575 Springer Krakow Poland September 2010

[12] M Albayrak and N Allahverdi ldquoDevelopment a new mu-tation operator to solve the traveling salesman problem by aidof genetic algorithmsrdquo Expert Systems with Applicationsvol 38 no 3 pp 1313ndash1320 2011

[13] G Dantzig R Fulkerson and S Johnson ldquoSolution of a large-scale traveling-salesman problemrdquo Journal of the OperationsResearch Society of America vol 2 no 4 pp 393ndash410 1954

[14] M Nazarahari E Khanmirza and S Doostie ldquoMulti-ob-jective multi-robot path planning in continuous environmentusing an enhanced genetic algorithmrdquo Expert Systems withApplications vol 115 pp 106ndash120 2019

[15] J Ni K Wang H Huang et al ldquoRobot path planning basedon an improved genetic algorithm with variable lengthchromosomerdquo in Proceedings of the 2016 12th InternationalConference on Natural Computation Fuzzy Systems andKnowledge Discovery (ICNC-FSKD) pp 145ndash149 ChangshaChina August 2016

[16] C-C Tsai H-C Huang and C-K Chan ldquoParallel elite ge-netic algorithm and its application to global path planning forautonomous robot navigationrdquo IEEE Transactions on In-dustrial Electronics vol 58 no 10 pp 4813ndash4821 2011

[17] M M Malakiyeh S Shojaee and S Hamzehei JavaranldquoInsight into an implicit time integration method based onBezier curve and third-order Bernstein basis function forstructural dynamicsrdquo Asian Journal of Civil Engineeringvol 19 no 1 pp 1ndash11 2018

[18] N Arana-Daniel A A Gallegos C Lopez-Franco et alldquoSmooth global and local path planning for mobile robotusing particle swarm optimization radial basis functionssplines and Bezier curvesrdquo in Proceedings of the 2014 IEEE

Computational Intelligence and Neuroscience 9

Congress on Evolutionary Computation (CEC) pp 175ndash182Beijing China July 2014

[19] A A Neto DGMacharet andM FM Campos ldquoFeasible RRT-based path planning using seventh order Bezier curvesrdquo in Pro-ceedings of the IEEE International Conference on Intelligent Robotsand Systems pp 1445ndash1450 Taipei Taiwan October 2010

[20] B Song G Tian and F Zhou ldquoA comparison study on pathsmoothing algorithms for laser robot navigated mobile robotpath planning in intelligent spacerdquo Journal of Information andComputational Science vol 7 no 1 pp 2943ndash2950 2010

[21] K G Jolly R Sreerama Kumar and R Vijayakumar ldquoA Beziercurve based path planning in a multi-agent robot soccer systemwithout violating the acceleration limitsrdquo Robotics and Au-tonomous Systems vol 57 no 1 pp 23ndash33 2009

[22] Y J Ho and J S Liu ldquoCollision-free curvature-boundedsmooth path planning using composite Bezier curve based onVoronoi diagramrdquo in Proceedings of the 2009 IEEE Inter-national Symposium on Computational Intelligence in Roboticsand Automation-(CIRA) pp 463ndash468 Daejeon South KoreaDecember 2009

[23] M Elhoseny A arwat and A E Hassanien ldquoBezier curvebased path planning in a dynamic field using modified geneticalgorithmrdquo Journal of Computational Science vol 25pp 339ndash350 2018

[24] A arwat M Elhoseny A E Hassanien et al ldquoIntelligentBezier curve-based path planningmodel using chaotic particleswarm optimization algorithmrdquo Cluster Computing vol 22no S2 pp 4745ndash4766 2019

[25] L Han H Yashiro H T N Nejad et al ldquoBezier curve basedpath planning for autonomous vehicle in urban environ-mentrdquo in Proceedings of the 2010 IEEE Intelligent VehiclesSymposium pp 1036ndash1042 San Diego CA USA June 2010

[26] I Skrjanc and G Klancar ldquoOptimal cooperative collisionavoidance betweenmultiple robots based on BernsteinndashBeziercurvesrdquo Robotics and Autonomous Systems vol 58 no 1pp 1ndash9 2010

[27] K Yang and S Sukkarieh ldquoAn analytical continuous-cur-vature path-smoothing algorithmrdquo IEEE Transactions onRobotics vol 26 no 3 pp 561ndash568 2010

[28] S H Chia K L Su J H Guo et al ldquoAnt colony system basedmobile robot path planningrdquo in Proceedings of the 2010 FourthInternational Conference on Genetic and Evolutionary Com-puting pp 210ndash213 Shenzhen China December 2010

[29] A T Ismail A Sheta and M Al-Weshah ldquoA mobile robotpath planning using genetic algorithm in static environmentrdquoJournal of Computer Science vol 4 no 4 pp 341ndash344 2008

[30] I Chaari A Koubaa H Bennaceur et al ldquoSmartPATH ahybrid ACO-GA algorithm for robot path planningrdquo inProceedings of the 2012 IEEE Congress on EvolutionaryComputation pp 1ndash8 Brisbane QLD Australia June 2012

[31] K Liang Z J Chen and X Q Yan ldquoMobile robot pathplanning simulation researchrdquoModern Electronic Technologyvol 41 no 17 pp 167ndash172 2018

[32] MMMalakiyeh S Shojaee and S H Javaran ldquoDevelopmentof a direct time integration method based on Bezier curve and5th-order Bernstein basis functionrdquo Computers amp Structuresvol 194 pp 15ndash31 2018

[33] N Wang H Guo B Chen et al ldquoDesign of a rotary dielectricelastomer actuator using a topology optimization methodbased on pairs of curvesrdquo Smart Materials and Structuresvol 27 no 5 Article ID 055011 2018

[34] B Song Z Wang L Zou L Xu and F E Alsaadi ldquoA newapproach to smooth global path planning of mobile robots

with kinematic constraintsrdquo International Journal of MachineLearning and Cybernetics vol 10 no 1 pp 107ndash119 2019

[35] B Song Z Wang L Zou et al ldquoOn global smooth pathplanning for mobile robots using a novel multimodal delayedPSO algorithmrdquo Cognitive Computation vol 9 no 1pp 5ndash17 2017

10 Computational Intelligence and Neuroscience

statistical results are given in Table 1 where ldquomdashrdquo means thatstatistical results cannot be obtained

Combining Figure 4 and Table 1 the following can befound in terms of path length (1) ACO the GA and theASFA-GA can plan a collision-free path (as shown inFigures 4(a) 4(b) and 4(d)) with the path lengths being346248 328663 and 312144 respectively Compared withthe BCA (299416) the planning paths of ACO the GA andthe ASFA-GA are longer which is caused by a lot of re-dundant nodes and redundant infection points in the path(2) When the GA is introduced into ACO ie for the ACO-GA the path length is 317964 this is an improved per-formance compared with just ACO and just the GAHowever there are still peak inflection points (egFigure 4(c)) (3) It can be seen from Figures 4(e) and 4(f)that the quality of the path obtained by the BCA and BCA-Q

has been significantly improved here the blue circles are thecontrol points of the Bezier curve and the red line is theoptimal smooth path e path lengths are 299416 and311843 respectively Combined with Table 1 the minimumand average values of the planned path lengths obtained byusing the BCA and BCA-Q are better than those of othermethods e reason for this is that the improved algorithmreduces redundant inflection points and redundant nodes inthe path planning therefore the path is smoother andshorter (4) Although the path length planned in Figure 4(f )is longer than that shown in Figure 4(e) the mobile securityof the robot is guaranteed and the path planning perfor-mance is better than that of other algorithms

In terms of running time Table 1 shows that theASFA-GA is the best algorithm in terms of required timefor simulations Moreover the ACO-GA has increased

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

Intersection (7 8)

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789

1011121314151617181920

Intersection (7 8)

(b)

Figure 2 Single-point crossover operation where intersection (7 8) is a crossover point (a) Control points before crossing (b) controlpoints after crossing

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(a)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

Figure 3 Path planning environments (a) environment 1 (b) environment 2

Computational Intelligence and Neuroscience 5

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(a)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(c)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(d)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(e)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(f )

Figure 4 Path planning simulation results in environment 1 (a) Path planned by ACO (b) path planned by the GA (c) path planned by theACO-GA (d) path planned by the ASFA-GA (e) path planned by the BCA (f ) path planned by the BCA-Q

6 Computational Intelligence and Neuroscience

Table 1 Path planning statistics in environment 1

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 346248 503794 425416 104718 204864 156471GA 328663 386874 357144 504718 701453 614772ACO-GA 317964 354147 334172 595674 844792 673358ASFA-GA 312144 335479 323648 84716 mdash mdashBCA 299416 324471 309436 1364794 2641479 2024861BCA-Q 311843 348517 329973 1746659 2963371 2344781

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

20

(a)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

20

(c)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(d)

Figure 5 Continued

Computational Intelligence and Neuroscience 7

simulation time due to the introduction of the GA is isbecause the GA introduces crossover and mutation op-erations in the process of path planning Finally Table 1shows that although the simulation times of the BCA andBCA-Q are longer than those of other algorithms theirminimum path lengths are shorter erefore comparedwith the traditional algorithms the proposed algorithmhas better performance in terms of path length andsmoothness in the environment with low real-timerequirements

42 Effectiveness Experiment of Bezier Smoothing AlgorithmsIn order to verify the performance of the proposed algo-rithm a complex grid environment of 20 times 20 is establishedin this paper e algorithms shown in Section 41 are ap-plied in environment 2 Figure 5 shows the path planningresults with different algorithms In order to eliminate theinfluence of random and other contingency factors on thealgorithm the aforementioned algorithms are executed 30times independently and the statistical results are recordedin Table 2 where ldquomdashrdquomeans that statistical results cannot beobtained

Based on the analysis of Figure 5 and Table 2 it can beobserved that the obstacle coverage of environment 2 is

increased by 575 compared with environment 1However the added coverage makes it more difficult tosearch for the global optimal solution Compared withTable 1 the simulation time and distance of path planningin Table 2 are increasede path length of the ASFA-GA is323821 Although the path is not the longest of all al-gorithms there are obvious inflection points (seeFigure 5(d)) (2) From Figures 5(a)ndash5(d) and Table 2 onecan see that ACO and the ACO-GA can also find a col-lision-free path in complex environments but their av-erage path length is longer compared to the ASFA-GAwhich is 336631 We can also observe from Figure 5(e)that the path obtained by the BCA is smoother and shortercompared with those obtained by other algorithms Re-dundant nodes are avoided and the path length is 303458From Table 2 we can see that the BCA is slightly insuf-ficient compared to other algorithms in terms of simu-lation time consumption However it is still effective incomplex environments from the perspective of path lengthand smoothness Additionally although the inflectionpoint disappears in the path planned by the BCA the pathis very close to the obstacle By increasing the safetydistance in the fitness function the BCA-Q improves thesecurity of the mobile robot which can be verified inFigure 5(f ) However this advantage is achieved by

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(e)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(f )

Figure 5 Path planning simulation results in environment 2 (a) Path planned by ACO (b) path planned by the GA (c) path planned by theACO-GA (d) path planned by the ASFA-GA (e) path planned by the BCA (f ) path planned by the BCA-Q

Table 2 Path planning statistics in environment 2

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 354526 561147 459981 257242 387413 325469GA 337963 404483 375448 857157 1076814 961479ACO-GA 329688 373691 351472 1084743 1364731 1211173ASFA-GA 323821 347728 336631 204473 mdash mdashBCA 303458 337694 321473 2027781 3789283 2934417BCA-Q 318503 354473 337628 2865575 4026491 3454799

8 Computational Intelligence and Neuroscience

sacrificing the path length e planned path length of theBCA-Q is 318503 which is 496 greater than that of theBCA It is noteworthy that the path length of BCA-Q is stillshorter than those of ACO and the GA Moreover itssmoothness and path security are greatly improved

5 Conclusions

In this paper a new smooth path planning method com-bining genetic operators and Bezier curves has been pro-posed for mobile robots First the control points of theBezier curve were determined by a genetic operation and thesmoothing characteristics of the Bezier curve were used tomake the planning path smoother and more consistentwhich reduced the energy loss in robot movement Secondthe safety distance was added to the fitness function andcould be dynamically adjusted according to the distancebetween the path and the obstacle to ensure the safe andefficient movement of the robot Finally the simulationresults showed that the proposed algorithm was effective infinding an optimal path this optimal path was shortersmoother and safer than those obtained by traditional al-gorithms However the simulation time was increased iswas because the improved algorithm paid more attention tothe quality of the solution in the process of path planningere still exist various meaningful topics to be addressedsuch as how to select the most appropriate number ofcontrol points of the continuous Bezier curve how to applythe algorithm to more complex practical environments(such as rooms or other special environments) and how toimprove the calculation efficiency with more control points

Data Availability

e data used to support the findings of this study are in-cluded within the article All required models and param-eters are listed in the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported in part by the National Key Re-search and Development Program of China under Grant2016YFE0104600 and the Science and Technology Programof Henan Province of China under Grant 172102410071

References

[1] M A P Garcia O Montiel O Castillo R Sepulveda andPMelin ldquoPath planning for autonomousmobile robot navigationwith ant colony optimization and fuzzy cost function evaluationrdquoApplied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[2] G Li andW Chou ldquoPath planning for mobile robot using self-adaptive learning particle swarm optimizationrdquo Science ChinaInformation Sciences vol 61 no 5 Article ID 052204 2018

[3] R K Mandava S Bondada and P R Vundavilli ldquoAn op-timized path planning for the mobile robot using potential

field method and PSO algorithmrdquo Advances in IntelligentSystems and Computing vol 8 no 7 pp 139ndash150 2019

[4] R Li W Wu and H Qiao ldquoe compliance of robotichandsmdashfrom functionality to mechanismrdquo Assembly Auto-mation vol 35 no 3 pp 281ndash286 2015

[5] D C Robinson D A Sanders and EMazharsolook ldquoAmbientintelligence for optimal manufacturing and energy efficiencyrdquoAssembly Automation vol 35 no 3 pp 234ndash248 2015

[6] B Song Z Wang and L Sheng ldquoA new genetic algorithmapproach to smooth path planning for mobile robotsrdquo As-sembly Automation vol 36 no 2 pp 138ndash145 2016

[7] A Bakdi A Hentout H Boutami A Maoudj O Hachourand B Bouzouia ldquoOptimal path planning and execution formobile robots using genetic algorithm and adaptive fuzzy-logic controlrdquo Robotics and Autonomous Systems vol 89pp 95ndash109 2017

[8] Y Deng Y Liu and D Zhou ldquoAn improved genetic algo-rithm with initial population strategy for symmetric TSPrdquoMathematical Problems in Engineering vol 2015 Article ID212794 6 pages 2015

[9] B K Patle D R K Parhi A Jagadeesh and S K KashyapldquoMatrix-binary codes based genetic algorithm for pathplanning of mobile robotrdquo Computers amp Electrical Engi-neering vol 67 pp 708ndash728 2018

[10] M A Contreras-Cruz V Ayala-Ramirez and U H Hernandez-Belmonte ldquoMobile robot path planning using artificial beecolony and evolutionary programmingrdquo Applied Soft Comput-ing vol 30 pp 319ndash328 2015

[11] D Whitley D Hains and A Howe ldquoA hybrid genetic al-gorithm for the traveling salesman problem using generalizedpartition crossoverrdquo in Proceedings of the InternationalConference on Parallel Problem Solving from Naturepp 566ndash575 Springer Krakow Poland September 2010

[12] M Albayrak and N Allahverdi ldquoDevelopment a new mu-tation operator to solve the traveling salesman problem by aidof genetic algorithmsrdquo Expert Systems with Applicationsvol 38 no 3 pp 1313ndash1320 2011

[13] G Dantzig R Fulkerson and S Johnson ldquoSolution of a large-scale traveling-salesman problemrdquo Journal of the OperationsResearch Society of America vol 2 no 4 pp 393ndash410 1954

[14] M Nazarahari E Khanmirza and S Doostie ldquoMulti-ob-jective multi-robot path planning in continuous environmentusing an enhanced genetic algorithmrdquo Expert Systems withApplications vol 115 pp 106ndash120 2019

[15] J Ni K Wang H Huang et al ldquoRobot path planning basedon an improved genetic algorithm with variable lengthchromosomerdquo in Proceedings of the 2016 12th InternationalConference on Natural Computation Fuzzy Systems andKnowledge Discovery (ICNC-FSKD) pp 145ndash149 ChangshaChina August 2016

[16] C-C Tsai H-C Huang and C-K Chan ldquoParallel elite ge-netic algorithm and its application to global path planning forautonomous robot navigationrdquo IEEE Transactions on In-dustrial Electronics vol 58 no 10 pp 4813ndash4821 2011

[17] M M Malakiyeh S Shojaee and S Hamzehei JavaranldquoInsight into an implicit time integration method based onBezier curve and third-order Bernstein basis function forstructural dynamicsrdquo Asian Journal of Civil Engineeringvol 19 no 1 pp 1ndash11 2018

[18] N Arana-Daniel A A Gallegos C Lopez-Franco et alldquoSmooth global and local path planning for mobile robotusing particle swarm optimization radial basis functionssplines and Bezier curvesrdquo in Proceedings of the 2014 IEEE

Computational Intelligence and Neuroscience 9

Congress on Evolutionary Computation (CEC) pp 175ndash182Beijing China July 2014

[19] A A Neto DGMacharet andM FM Campos ldquoFeasible RRT-based path planning using seventh order Bezier curvesrdquo in Pro-ceedings of the IEEE International Conference on Intelligent Robotsand Systems pp 1445ndash1450 Taipei Taiwan October 2010

[20] B Song G Tian and F Zhou ldquoA comparison study on pathsmoothing algorithms for laser robot navigated mobile robotpath planning in intelligent spacerdquo Journal of Information andComputational Science vol 7 no 1 pp 2943ndash2950 2010

[21] K G Jolly R Sreerama Kumar and R Vijayakumar ldquoA Beziercurve based path planning in a multi-agent robot soccer systemwithout violating the acceleration limitsrdquo Robotics and Au-tonomous Systems vol 57 no 1 pp 23ndash33 2009

[22] Y J Ho and J S Liu ldquoCollision-free curvature-boundedsmooth path planning using composite Bezier curve based onVoronoi diagramrdquo in Proceedings of the 2009 IEEE Inter-national Symposium on Computational Intelligence in Roboticsand Automation-(CIRA) pp 463ndash468 Daejeon South KoreaDecember 2009

[23] M Elhoseny A arwat and A E Hassanien ldquoBezier curvebased path planning in a dynamic field using modified geneticalgorithmrdquo Journal of Computational Science vol 25pp 339ndash350 2018

[24] A arwat M Elhoseny A E Hassanien et al ldquoIntelligentBezier curve-based path planningmodel using chaotic particleswarm optimization algorithmrdquo Cluster Computing vol 22no S2 pp 4745ndash4766 2019

[25] L Han H Yashiro H T N Nejad et al ldquoBezier curve basedpath planning for autonomous vehicle in urban environ-mentrdquo in Proceedings of the 2010 IEEE Intelligent VehiclesSymposium pp 1036ndash1042 San Diego CA USA June 2010

[26] I Skrjanc and G Klancar ldquoOptimal cooperative collisionavoidance betweenmultiple robots based on BernsteinndashBeziercurvesrdquo Robotics and Autonomous Systems vol 58 no 1pp 1ndash9 2010

[27] K Yang and S Sukkarieh ldquoAn analytical continuous-cur-vature path-smoothing algorithmrdquo IEEE Transactions onRobotics vol 26 no 3 pp 561ndash568 2010

[28] S H Chia K L Su J H Guo et al ldquoAnt colony system basedmobile robot path planningrdquo in Proceedings of the 2010 FourthInternational Conference on Genetic and Evolutionary Com-puting pp 210ndash213 Shenzhen China December 2010

[29] A T Ismail A Sheta and M Al-Weshah ldquoA mobile robotpath planning using genetic algorithm in static environmentrdquoJournal of Computer Science vol 4 no 4 pp 341ndash344 2008

[30] I Chaari A Koubaa H Bennaceur et al ldquoSmartPATH ahybrid ACO-GA algorithm for robot path planningrdquo inProceedings of the 2012 IEEE Congress on EvolutionaryComputation pp 1ndash8 Brisbane QLD Australia June 2012

[31] K Liang Z J Chen and X Q Yan ldquoMobile robot pathplanning simulation researchrdquoModern Electronic Technologyvol 41 no 17 pp 167ndash172 2018

[32] MMMalakiyeh S Shojaee and S H Javaran ldquoDevelopmentof a direct time integration method based on Bezier curve and5th-order Bernstein basis functionrdquo Computers amp Structuresvol 194 pp 15ndash31 2018

[33] N Wang H Guo B Chen et al ldquoDesign of a rotary dielectricelastomer actuator using a topology optimization methodbased on pairs of curvesrdquo Smart Materials and Structuresvol 27 no 5 Article ID 055011 2018

[34] B Song Z Wang L Zou L Xu and F E Alsaadi ldquoA newapproach to smooth global path planning of mobile robots

with kinematic constraintsrdquo International Journal of MachineLearning and Cybernetics vol 10 no 1 pp 107ndash119 2019

[35] B Song Z Wang L Zou et al ldquoOn global smooth pathplanning for mobile robots using a novel multimodal delayedPSO algorithmrdquo Cognitive Computation vol 9 no 1pp 5ndash17 2017

10 Computational Intelligence and Neuroscience

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(a)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(c)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(d)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(e)

0 2 4 15 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(f )

Figure 4 Path planning simulation results in environment 1 (a) Path planned by ACO (b) path planned by the GA (c) path planned by theACO-GA (d) path planned by the ASFA-GA (e) path planned by the BCA (f ) path planned by the BCA-Q

6 Computational Intelligence and Neuroscience

Table 1 Path planning statistics in environment 1

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 346248 503794 425416 104718 204864 156471GA 328663 386874 357144 504718 701453 614772ACO-GA 317964 354147 334172 595674 844792 673358ASFA-GA 312144 335479 323648 84716 mdash mdashBCA 299416 324471 309436 1364794 2641479 2024861BCA-Q 311843 348517 329973 1746659 2963371 2344781

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

20

(a)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

20

(c)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(d)

Figure 5 Continued

Computational Intelligence and Neuroscience 7

simulation time due to the introduction of the GA is isbecause the GA introduces crossover and mutation op-erations in the process of path planning Finally Table 1shows that although the simulation times of the BCA andBCA-Q are longer than those of other algorithms theirminimum path lengths are shorter erefore comparedwith the traditional algorithms the proposed algorithmhas better performance in terms of path length andsmoothness in the environment with low real-timerequirements

42 Effectiveness Experiment of Bezier Smoothing AlgorithmsIn order to verify the performance of the proposed algo-rithm a complex grid environment of 20 times 20 is establishedin this paper e algorithms shown in Section 41 are ap-plied in environment 2 Figure 5 shows the path planningresults with different algorithms In order to eliminate theinfluence of random and other contingency factors on thealgorithm the aforementioned algorithms are executed 30times independently and the statistical results are recordedin Table 2 where ldquomdashrdquomeans that statistical results cannot beobtained

Based on the analysis of Figure 5 and Table 2 it can beobserved that the obstacle coverage of environment 2 is

increased by 575 compared with environment 1However the added coverage makes it more difficult tosearch for the global optimal solution Compared withTable 1 the simulation time and distance of path planningin Table 2 are increasede path length of the ASFA-GA is323821 Although the path is not the longest of all al-gorithms there are obvious inflection points (seeFigure 5(d)) (2) From Figures 5(a)ndash5(d) and Table 2 onecan see that ACO and the ACO-GA can also find a col-lision-free path in complex environments but their av-erage path length is longer compared to the ASFA-GAwhich is 336631 We can also observe from Figure 5(e)that the path obtained by the BCA is smoother and shortercompared with those obtained by other algorithms Re-dundant nodes are avoided and the path length is 303458From Table 2 we can see that the BCA is slightly insuf-ficient compared to other algorithms in terms of simu-lation time consumption However it is still effective incomplex environments from the perspective of path lengthand smoothness Additionally although the inflectionpoint disappears in the path planned by the BCA the pathis very close to the obstacle By increasing the safetydistance in the fitness function the BCA-Q improves thesecurity of the mobile robot which can be verified inFigure 5(f ) However this advantage is achieved by

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(e)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(f )

Figure 5 Path planning simulation results in environment 2 (a) Path planned by ACO (b) path planned by the GA (c) path planned by theACO-GA (d) path planned by the ASFA-GA (e) path planned by the BCA (f ) path planned by the BCA-Q

Table 2 Path planning statistics in environment 2

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 354526 561147 459981 257242 387413 325469GA 337963 404483 375448 857157 1076814 961479ACO-GA 329688 373691 351472 1084743 1364731 1211173ASFA-GA 323821 347728 336631 204473 mdash mdashBCA 303458 337694 321473 2027781 3789283 2934417BCA-Q 318503 354473 337628 2865575 4026491 3454799

8 Computational Intelligence and Neuroscience

sacrificing the path length e planned path length of theBCA-Q is 318503 which is 496 greater than that of theBCA It is noteworthy that the path length of BCA-Q is stillshorter than those of ACO and the GA Moreover itssmoothness and path security are greatly improved

5 Conclusions

In this paper a new smooth path planning method com-bining genetic operators and Bezier curves has been pro-posed for mobile robots First the control points of theBezier curve were determined by a genetic operation and thesmoothing characteristics of the Bezier curve were used tomake the planning path smoother and more consistentwhich reduced the energy loss in robot movement Secondthe safety distance was added to the fitness function andcould be dynamically adjusted according to the distancebetween the path and the obstacle to ensure the safe andefficient movement of the robot Finally the simulationresults showed that the proposed algorithm was effective infinding an optimal path this optimal path was shortersmoother and safer than those obtained by traditional al-gorithms However the simulation time was increased iswas because the improved algorithm paid more attention tothe quality of the solution in the process of path planningere still exist various meaningful topics to be addressedsuch as how to select the most appropriate number ofcontrol points of the continuous Bezier curve how to applythe algorithm to more complex practical environments(such as rooms or other special environments) and how toimprove the calculation efficiency with more control points

Data Availability

e data used to support the findings of this study are in-cluded within the article All required models and param-eters are listed in the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported in part by the National Key Re-search and Development Program of China under Grant2016YFE0104600 and the Science and Technology Programof Henan Province of China under Grant 172102410071

References

[1] M A P Garcia O Montiel O Castillo R Sepulveda andPMelin ldquoPath planning for autonomousmobile robot navigationwith ant colony optimization and fuzzy cost function evaluationrdquoApplied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[2] G Li andW Chou ldquoPath planning for mobile robot using self-adaptive learning particle swarm optimizationrdquo Science ChinaInformation Sciences vol 61 no 5 Article ID 052204 2018

[3] R K Mandava S Bondada and P R Vundavilli ldquoAn op-timized path planning for the mobile robot using potential

field method and PSO algorithmrdquo Advances in IntelligentSystems and Computing vol 8 no 7 pp 139ndash150 2019

[4] R Li W Wu and H Qiao ldquoe compliance of robotichandsmdashfrom functionality to mechanismrdquo Assembly Auto-mation vol 35 no 3 pp 281ndash286 2015

[5] D C Robinson D A Sanders and EMazharsolook ldquoAmbientintelligence for optimal manufacturing and energy efficiencyrdquoAssembly Automation vol 35 no 3 pp 234ndash248 2015

[6] B Song Z Wang and L Sheng ldquoA new genetic algorithmapproach to smooth path planning for mobile robotsrdquo As-sembly Automation vol 36 no 2 pp 138ndash145 2016

[7] A Bakdi A Hentout H Boutami A Maoudj O Hachourand B Bouzouia ldquoOptimal path planning and execution formobile robots using genetic algorithm and adaptive fuzzy-logic controlrdquo Robotics and Autonomous Systems vol 89pp 95ndash109 2017

[8] Y Deng Y Liu and D Zhou ldquoAn improved genetic algo-rithm with initial population strategy for symmetric TSPrdquoMathematical Problems in Engineering vol 2015 Article ID212794 6 pages 2015

[9] B K Patle D R K Parhi A Jagadeesh and S K KashyapldquoMatrix-binary codes based genetic algorithm for pathplanning of mobile robotrdquo Computers amp Electrical Engi-neering vol 67 pp 708ndash728 2018

[10] M A Contreras-Cruz V Ayala-Ramirez and U H Hernandez-Belmonte ldquoMobile robot path planning using artificial beecolony and evolutionary programmingrdquo Applied Soft Comput-ing vol 30 pp 319ndash328 2015

[11] D Whitley D Hains and A Howe ldquoA hybrid genetic al-gorithm for the traveling salesman problem using generalizedpartition crossoverrdquo in Proceedings of the InternationalConference on Parallel Problem Solving from Naturepp 566ndash575 Springer Krakow Poland September 2010

[12] M Albayrak and N Allahverdi ldquoDevelopment a new mu-tation operator to solve the traveling salesman problem by aidof genetic algorithmsrdquo Expert Systems with Applicationsvol 38 no 3 pp 1313ndash1320 2011

[13] G Dantzig R Fulkerson and S Johnson ldquoSolution of a large-scale traveling-salesman problemrdquo Journal of the OperationsResearch Society of America vol 2 no 4 pp 393ndash410 1954

[14] M Nazarahari E Khanmirza and S Doostie ldquoMulti-ob-jective multi-robot path planning in continuous environmentusing an enhanced genetic algorithmrdquo Expert Systems withApplications vol 115 pp 106ndash120 2019

[15] J Ni K Wang H Huang et al ldquoRobot path planning basedon an improved genetic algorithm with variable lengthchromosomerdquo in Proceedings of the 2016 12th InternationalConference on Natural Computation Fuzzy Systems andKnowledge Discovery (ICNC-FSKD) pp 145ndash149 ChangshaChina August 2016

[16] C-C Tsai H-C Huang and C-K Chan ldquoParallel elite ge-netic algorithm and its application to global path planning forautonomous robot navigationrdquo IEEE Transactions on In-dustrial Electronics vol 58 no 10 pp 4813ndash4821 2011

[17] M M Malakiyeh S Shojaee and S Hamzehei JavaranldquoInsight into an implicit time integration method based onBezier curve and third-order Bernstein basis function forstructural dynamicsrdquo Asian Journal of Civil Engineeringvol 19 no 1 pp 1ndash11 2018

[18] N Arana-Daniel A A Gallegos C Lopez-Franco et alldquoSmooth global and local path planning for mobile robotusing particle swarm optimization radial basis functionssplines and Bezier curvesrdquo in Proceedings of the 2014 IEEE

Computational Intelligence and Neuroscience 9

Congress on Evolutionary Computation (CEC) pp 175ndash182Beijing China July 2014

[19] A A Neto DGMacharet andM FM Campos ldquoFeasible RRT-based path planning using seventh order Bezier curvesrdquo in Pro-ceedings of the IEEE International Conference on Intelligent Robotsand Systems pp 1445ndash1450 Taipei Taiwan October 2010

[20] B Song G Tian and F Zhou ldquoA comparison study on pathsmoothing algorithms for laser robot navigated mobile robotpath planning in intelligent spacerdquo Journal of Information andComputational Science vol 7 no 1 pp 2943ndash2950 2010

[21] K G Jolly R Sreerama Kumar and R Vijayakumar ldquoA Beziercurve based path planning in a multi-agent robot soccer systemwithout violating the acceleration limitsrdquo Robotics and Au-tonomous Systems vol 57 no 1 pp 23ndash33 2009

[22] Y J Ho and J S Liu ldquoCollision-free curvature-boundedsmooth path planning using composite Bezier curve based onVoronoi diagramrdquo in Proceedings of the 2009 IEEE Inter-national Symposium on Computational Intelligence in Roboticsand Automation-(CIRA) pp 463ndash468 Daejeon South KoreaDecember 2009

[23] M Elhoseny A arwat and A E Hassanien ldquoBezier curvebased path planning in a dynamic field using modified geneticalgorithmrdquo Journal of Computational Science vol 25pp 339ndash350 2018

[24] A arwat M Elhoseny A E Hassanien et al ldquoIntelligentBezier curve-based path planningmodel using chaotic particleswarm optimization algorithmrdquo Cluster Computing vol 22no S2 pp 4745ndash4766 2019

[25] L Han H Yashiro H T N Nejad et al ldquoBezier curve basedpath planning for autonomous vehicle in urban environ-mentrdquo in Proceedings of the 2010 IEEE Intelligent VehiclesSymposium pp 1036ndash1042 San Diego CA USA June 2010

[26] I Skrjanc and G Klancar ldquoOptimal cooperative collisionavoidance betweenmultiple robots based on BernsteinndashBeziercurvesrdquo Robotics and Autonomous Systems vol 58 no 1pp 1ndash9 2010

[27] K Yang and S Sukkarieh ldquoAn analytical continuous-cur-vature path-smoothing algorithmrdquo IEEE Transactions onRobotics vol 26 no 3 pp 561ndash568 2010

[28] S H Chia K L Su J H Guo et al ldquoAnt colony system basedmobile robot path planningrdquo in Proceedings of the 2010 FourthInternational Conference on Genetic and Evolutionary Com-puting pp 210ndash213 Shenzhen China December 2010

[29] A T Ismail A Sheta and M Al-Weshah ldquoA mobile robotpath planning using genetic algorithm in static environmentrdquoJournal of Computer Science vol 4 no 4 pp 341ndash344 2008

[30] I Chaari A Koubaa H Bennaceur et al ldquoSmartPATH ahybrid ACO-GA algorithm for robot path planningrdquo inProceedings of the 2012 IEEE Congress on EvolutionaryComputation pp 1ndash8 Brisbane QLD Australia June 2012

[31] K Liang Z J Chen and X Q Yan ldquoMobile robot pathplanning simulation researchrdquoModern Electronic Technologyvol 41 no 17 pp 167ndash172 2018

[32] MMMalakiyeh S Shojaee and S H Javaran ldquoDevelopmentof a direct time integration method based on Bezier curve and5th-order Bernstein basis functionrdquo Computers amp Structuresvol 194 pp 15ndash31 2018

[33] N Wang H Guo B Chen et al ldquoDesign of a rotary dielectricelastomer actuator using a topology optimization methodbased on pairs of curvesrdquo Smart Materials and Structuresvol 27 no 5 Article ID 055011 2018

[34] B Song Z Wang L Zou L Xu and F E Alsaadi ldquoA newapproach to smooth global path planning of mobile robots

with kinematic constraintsrdquo International Journal of MachineLearning and Cybernetics vol 10 no 1 pp 107ndash119 2019

[35] B Song Z Wang L Zou et al ldquoOn global smooth pathplanning for mobile robots using a novel multimodal delayedPSO algorithmrdquo Cognitive Computation vol 9 no 1pp 5ndash17 2017

10 Computational Intelligence and Neuroscience

Table 1 Path planning statistics in environment 1

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 346248 503794 425416 104718 204864 156471GA 328663 386874 357144 504718 701453 614772ACO-GA 317964 354147 334172 595674 844792 673358ASFA-GA 312144 335479 323648 84716 mdash mdashBCA 299416 324471 309436 1364794 2641479 2024861BCA-Q 311843 348517 329973 1746659 2963371 2344781

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

20

(a)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(b)

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

20

(c)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(d)

Figure 5 Continued

Computational Intelligence and Neuroscience 7

simulation time due to the introduction of the GA is isbecause the GA introduces crossover and mutation op-erations in the process of path planning Finally Table 1shows that although the simulation times of the BCA andBCA-Q are longer than those of other algorithms theirminimum path lengths are shorter erefore comparedwith the traditional algorithms the proposed algorithmhas better performance in terms of path length andsmoothness in the environment with low real-timerequirements

42 Effectiveness Experiment of Bezier Smoothing AlgorithmsIn order to verify the performance of the proposed algo-rithm a complex grid environment of 20 times 20 is establishedin this paper e algorithms shown in Section 41 are ap-plied in environment 2 Figure 5 shows the path planningresults with different algorithms In order to eliminate theinfluence of random and other contingency factors on thealgorithm the aforementioned algorithms are executed 30times independently and the statistical results are recordedin Table 2 where ldquomdashrdquomeans that statistical results cannot beobtained

Based on the analysis of Figure 5 and Table 2 it can beobserved that the obstacle coverage of environment 2 is

increased by 575 compared with environment 1However the added coverage makes it more difficult tosearch for the global optimal solution Compared withTable 1 the simulation time and distance of path planningin Table 2 are increasede path length of the ASFA-GA is323821 Although the path is not the longest of all al-gorithms there are obvious inflection points (seeFigure 5(d)) (2) From Figures 5(a)ndash5(d) and Table 2 onecan see that ACO and the ACO-GA can also find a col-lision-free path in complex environments but their av-erage path length is longer compared to the ASFA-GAwhich is 336631 We can also observe from Figure 5(e)that the path obtained by the BCA is smoother and shortercompared with those obtained by other algorithms Re-dundant nodes are avoided and the path length is 303458From Table 2 we can see that the BCA is slightly insuf-ficient compared to other algorithms in terms of simu-lation time consumption However it is still effective incomplex environments from the perspective of path lengthand smoothness Additionally although the inflectionpoint disappears in the path planned by the BCA the pathis very close to the obstacle By increasing the safetydistance in the fitness function the BCA-Q improves thesecurity of the mobile robot which can be verified inFigure 5(f ) However this advantage is achieved by

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(e)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(f )

Figure 5 Path planning simulation results in environment 2 (a) Path planned by ACO (b) path planned by the GA (c) path planned by theACO-GA (d) path planned by the ASFA-GA (e) path planned by the BCA (f ) path planned by the BCA-Q

Table 2 Path planning statistics in environment 2

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 354526 561147 459981 257242 387413 325469GA 337963 404483 375448 857157 1076814 961479ACO-GA 329688 373691 351472 1084743 1364731 1211173ASFA-GA 323821 347728 336631 204473 mdash mdashBCA 303458 337694 321473 2027781 3789283 2934417BCA-Q 318503 354473 337628 2865575 4026491 3454799

8 Computational Intelligence and Neuroscience

sacrificing the path length e planned path length of theBCA-Q is 318503 which is 496 greater than that of theBCA It is noteworthy that the path length of BCA-Q is stillshorter than those of ACO and the GA Moreover itssmoothness and path security are greatly improved

5 Conclusions

In this paper a new smooth path planning method com-bining genetic operators and Bezier curves has been pro-posed for mobile robots First the control points of theBezier curve were determined by a genetic operation and thesmoothing characteristics of the Bezier curve were used tomake the planning path smoother and more consistentwhich reduced the energy loss in robot movement Secondthe safety distance was added to the fitness function andcould be dynamically adjusted according to the distancebetween the path and the obstacle to ensure the safe andefficient movement of the robot Finally the simulationresults showed that the proposed algorithm was effective infinding an optimal path this optimal path was shortersmoother and safer than those obtained by traditional al-gorithms However the simulation time was increased iswas because the improved algorithm paid more attention tothe quality of the solution in the process of path planningere still exist various meaningful topics to be addressedsuch as how to select the most appropriate number ofcontrol points of the continuous Bezier curve how to applythe algorithm to more complex practical environments(such as rooms or other special environments) and how toimprove the calculation efficiency with more control points

Data Availability

e data used to support the findings of this study are in-cluded within the article All required models and param-eters are listed in the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported in part by the National Key Re-search and Development Program of China under Grant2016YFE0104600 and the Science and Technology Programof Henan Province of China under Grant 172102410071

References

[1] M A P Garcia O Montiel O Castillo R Sepulveda andPMelin ldquoPath planning for autonomousmobile robot navigationwith ant colony optimization and fuzzy cost function evaluationrdquoApplied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[2] G Li andW Chou ldquoPath planning for mobile robot using self-adaptive learning particle swarm optimizationrdquo Science ChinaInformation Sciences vol 61 no 5 Article ID 052204 2018

[3] R K Mandava S Bondada and P R Vundavilli ldquoAn op-timized path planning for the mobile robot using potential

field method and PSO algorithmrdquo Advances in IntelligentSystems and Computing vol 8 no 7 pp 139ndash150 2019

[4] R Li W Wu and H Qiao ldquoe compliance of robotichandsmdashfrom functionality to mechanismrdquo Assembly Auto-mation vol 35 no 3 pp 281ndash286 2015

[5] D C Robinson D A Sanders and EMazharsolook ldquoAmbientintelligence for optimal manufacturing and energy efficiencyrdquoAssembly Automation vol 35 no 3 pp 234ndash248 2015

[6] B Song Z Wang and L Sheng ldquoA new genetic algorithmapproach to smooth path planning for mobile robotsrdquo As-sembly Automation vol 36 no 2 pp 138ndash145 2016

[7] A Bakdi A Hentout H Boutami A Maoudj O Hachourand B Bouzouia ldquoOptimal path planning and execution formobile robots using genetic algorithm and adaptive fuzzy-logic controlrdquo Robotics and Autonomous Systems vol 89pp 95ndash109 2017

[8] Y Deng Y Liu and D Zhou ldquoAn improved genetic algo-rithm with initial population strategy for symmetric TSPrdquoMathematical Problems in Engineering vol 2015 Article ID212794 6 pages 2015

[9] B K Patle D R K Parhi A Jagadeesh and S K KashyapldquoMatrix-binary codes based genetic algorithm for pathplanning of mobile robotrdquo Computers amp Electrical Engi-neering vol 67 pp 708ndash728 2018

[10] M A Contreras-Cruz V Ayala-Ramirez and U H Hernandez-Belmonte ldquoMobile robot path planning using artificial beecolony and evolutionary programmingrdquo Applied Soft Comput-ing vol 30 pp 319ndash328 2015

[11] D Whitley D Hains and A Howe ldquoA hybrid genetic al-gorithm for the traveling salesman problem using generalizedpartition crossoverrdquo in Proceedings of the InternationalConference on Parallel Problem Solving from Naturepp 566ndash575 Springer Krakow Poland September 2010

[12] M Albayrak and N Allahverdi ldquoDevelopment a new mu-tation operator to solve the traveling salesman problem by aidof genetic algorithmsrdquo Expert Systems with Applicationsvol 38 no 3 pp 1313ndash1320 2011

[13] G Dantzig R Fulkerson and S Johnson ldquoSolution of a large-scale traveling-salesman problemrdquo Journal of the OperationsResearch Society of America vol 2 no 4 pp 393ndash410 1954

[14] M Nazarahari E Khanmirza and S Doostie ldquoMulti-ob-jective multi-robot path planning in continuous environmentusing an enhanced genetic algorithmrdquo Expert Systems withApplications vol 115 pp 106ndash120 2019

[15] J Ni K Wang H Huang et al ldquoRobot path planning basedon an improved genetic algorithm with variable lengthchromosomerdquo in Proceedings of the 2016 12th InternationalConference on Natural Computation Fuzzy Systems andKnowledge Discovery (ICNC-FSKD) pp 145ndash149 ChangshaChina August 2016

[16] C-C Tsai H-C Huang and C-K Chan ldquoParallel elite ge-netic algorithm and its application to global path planning forautonomous robot navigationrdquo IEEE Transactions on In-dustrial Electronics vol 58 no 10 pp 4813ndash4821 2011

[17] M M Malakiyeh S Shojaee and S Hamzehei JavaranldquoInsight into an implicit time integration method based onBezier curve and third-order Bernstein basis function forstructural dynamicsrdquo Asian Journal of Civil Engineeringvol 19 no 1 pp 1ndash11 2018

[18] N Arana-Daniel A A Gallegos C Lopez-Franco et alldquoSmooth global and local path planning for mobile robotusing particle swarm optimization radial basis functionssplines and Bezier curvesrdquo in Proceedings of the 2014 IEEE

Computational Intelligence and Neuroscience 9

Congress on Evolutionary Computation (CEC) pp 175ndash182Beijing China July 2014

[19] A A Neto DGMacharet andM FM Campos ldquoFeasible RRT-based path planning using seventh order Bezier curvesrdquo in Pro-ceedings of the IEEE International Conference on Intelligent Robotsand Systems pp 1445ndash1450 Taipei Taiwan October 2010

[20] B Song G Tian and F Zhou ldquoA comparison study on pathsmoothing algorithms for laser robot navigated mobile robotpath planning in intelligent spacerdquo Journal of Information andComputational Science vol 7 no 1 pp 2943ndash2950 2010

[21] K G Jolly R Sreerama Kumar and R Vijayakumar ldquoA Beziercurve based path planning in a multi-agent robot soccer systemwithout violating the acceleration limitsrdquo Robotics and Au-tonomous Systems vol 57 no 1 pp 23ndash33 2009

[22] Y J Ho and J S Liu ldquoCollision-free curvature-boundedsmooth path planning using composite Bezier curve based onVoronoi diagramrdquo in Proceedings of the 2009 IEEE Inter-national Symposium on Computational Intelligence in Roboticsand Automation-(CIRA) pp 463ndash468 Daejeon South KoreaDecember 2009

[23] M Elhoseny A arwat and A E Hassanien ldquoBezier curvebased path planning in a dynamic field using modified geneticalgorithmrdquo Journal of Computational Science vol 25pp 339ndash350 2018

[24] A arwat M Elhoseny A E Hassanien et al ldquoIntelligentBezier curve-based path planningmodel using chaotic particleswarm optimization algorithmrdquo Cluster Computing vol 22no S2 pp 4745ndash4766 2019

[25] L Han H Yashiro H T N Nejad et al ldquoBezier curve basedpath planning for autonomous vehicle in urban environ-mentrdquo in Proceedings of the 2010 IEEE Intelligent VehiclesSymposium pp 1036ndash1042 San Diego CA USA June 2010

[26] I Skrjanc and G Klancar ldquoOptimal cooperative collisionavoidance betweenmultiple robots based on BernsteinndashBeziercurvesrdquo Robotics and Autonomous Systems vol 58 no 1pp 1ndash9 2010

[27] K Yang and S Sukkarieh ldquoAn analytical continuous-cur-vature path-smoothing algorithmrdquo IEEE Transactions onRobotics vol 26 no 3 pp 561ndash568 2010

[28] S H Chia K L Su J H Guo et al ldquoAnt colony system basedmobile robot path planningrdquo in Proceedings of the 2010 FourthInternational Conference on Genetic and Evolutionary Com-puting pp 210ndash213 Shenzhen China December 2010

[29] A T Ismail A Sheta and M Al-Weshah ldquoA mobile robotpath planning using genetic algorithm in static environmentrdquoJournal of Computer Science vol 4 no 4 pp 341ndash344 2008

[30] I Chaari A Koubaa H Bennaceur et al ldquoSmartPATH ahybrid ACO-GA algorithm for robot path planningrdquo inProceedings of the 2012 IEEE Congress on EvolutionaryComputation pp 1ndash8 Brisbane QLD Australia June 2012

[31] K Liang Z J Chen and X Q Yan ldquoMobile robot pathplanning simulation researchrdquoModern Electronic Technologyvol 41 no 17 pp 167ndash172 2018

[32] MMMalakiyeh S Shojaee and S H Javaran ldquoDevelopmentof a direct time integration method based on Bezier curve and5th-order Bernstein basis functionrdquo Computers amp Structuresvol 194 pp 15ndash31 2018

[33] N Wang H Guo B Chen et al ldquoDesign of a rotary dielectricelastomer actuator using a topology optimization methodbased on pairs of curvesrdquo Smart Materials and Structuresvol 27 no 5 Article ID 055011 2018

[34] B Song Z Wang L Zou L Xu and F E Alsaadi ldquoA newapproach to smooth global path planning of mobile robots

with kinematic constraintsrdquo International Journal of MachineLearning and Cybernetics vol 10 no 1 pp 107ndash119 2019

[35] B Song Z Wang L Zou et al ldquoOn global smooth pathplanning for mobile robots using a novel multimodal delayedPSO algorithmrdquo Cognitive Computation vol 9 no 1pp 5ndash17 2017

10 Computational Intelligence and Neuroscience

simulation time due to the introduction of the GA is isbecause the GA introduces crossover and mutation op-erations in the process of path planning Finally Table 1shows that although the simulation times of the BCA andBCA-Q are longer than those of other algorithms theirminimum path lengths are shorter erefore comparedwith the traditional algorithms the proposed algorithmhas better performance in terms of path length andsmoothness in the environment with low real-timerequirements

42 Effectiveness Experiment of Bezier Smoothing AlgorithmsIn order to verify the performance of the proposed algo-rithm a complex grid environment of 20 times 20 is establishedin this paper e algorithms shown in Section 41 are ap-plied in environment 2 Figure 5 shows the path planningresults with different algorithms In order to eliminate theinfluence of random and other contingency factors on thealgorithm the aforementioned algorithms are executed 30times independently and the statistical results are recordedin Table 2 where ldquomdashrdquomeans that statistical results cannot beobtained

Based on the analysis of Figure 5 and Table 2 it can beobserved that the obstacle coverage of environment 2 is

increased by 575 compared with environment 1However the added coverage makes it more difficult tosearch for the global optimal solution Compared withTable 1 the simulation time and distance of path planningin Table 2 are increasede path length of the ASFA-GA is323821 Although the path is not the longest of all al-gorithms there are obvious inflection points (seeFigure 5(d)) (2) From Figures 5(a)ndash5(d) and Table 2 onecan see that ACO and the ACO-GA can also find a col-lision-free path in complex environments but their av-erage path length is longer compared to the ASFA-GAwhich is 336631 We can also observe from Figure 5(e)that the path obtained by the BCA is smoother and shortercompared with those obtained by other algorithms Re-dundant nodes are avoided and the path length is 303458From Table 2 we can see that the BCA is slightly insuf-ficient compared to other algorithms in terms of simu-lation time consumption However it is still effective incomplex environments from the perspective of path lengthand smoothness Additionally although the inflectionpoint disappears in the path planned by the BCA the pathis very close to the obstacle By increasing the safetydistance in the fitness function the BCA-Q improves thesecurity of the mobile robot which can be verified inFigure 5(f ) However this advantage is achieved by

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(e)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

(f )

Figure 5 Path planning simulation results in environment 2 (a) Path planned by ACO (b) path planned by the GA (c) path planned by theACO-GA (d) path planned by the ASFA-GA (e) path planned by the BCA (f ) path planned by the BCA-Q

Table 2 Path planning statistics in environment 2

Path planning algorithmPlanning path length Simulation time consumption (s)

Minimum Maximum Average Minimum Maximum AverageACO 354526 561147 459981 257242 387413 325469GA 337963 404483 375448 857157 1076814 961479ACO-GA 329688 373691 351472 1084743 1364731 1211173ASFA-GA 323821 347728 336631 204473 mdash mdashBCA 303458 337694 321473 2027781 3789283 2934417BCA-Q 318503 354473 337628 2865575 4026491 3454799

8 Computational Intelligence and Neuroscience

sacrificing the path length e planned path length of theBCA-Q is 318503 which is 496 greater than that of theBCA It is noteworthy that the path length of BCA-Q is stillshorter than those of ACO and the GA Moreover itssmoothness and path security are greatly improved

5 Conclusions

In this paper a new smooth path planning method com-bining genetic operators and Bezier curves has been pro-posed for mobile robots First the control points of theBezier curve were determined by a genetic operation and thesmoothing characteristics of the Bezier curve were used tomake the planning path smoother and more consistentwhich reduced the energy loss in robot movement Secondthe safety distance was added to the fitness function andcould be dynamically adjusted according to the distancebetween the path and the obstacle to ensure the safe andefficient movement of the robot Finally the simulationresults showed that the proposed algorithm was effective infinding an optimal path this optimal path was shortersmoother and safer than those obtained by traditional al-gorithms However the simulation time was increased iswas because the improved algorithm paid more attention tothe quality of the solution in the process of path planningere still exist various meaningful topics to be addressedsuch as how to select the most appropriate number ofcontrol points of the continuous Bezier curve how to applythe algorithm to more complex practical environments(such as rooms or other special environments) and how toimprove the calculation efficiency with more control points

Data Availability

e data used to support the findings of this study are in-cluded within the article All required models and param-eters are listed in the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported in part by the National Key Re-search and Development Program of China under Grant2016YFE0104600 and the Science and Technology Programof Henan Province of China under Grant 172102410071

References

[1] M A P Garcia O Montiel O Castillo R Sepulveda andPMelin ldquoPath planning for autonomousmobile robot navigationwith ant colony optimization and fuzzy cost function evaluationrdquoApplied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[2] G Li andW Chou ldquoPath planning for mobile robot using self-adaptive learning particle swarm optimizationrdquo Science ChinaInformation Sciences vol 61 no 5 Article ID 052204 2018

[3] R K Mandava S Bondada and P R Vundavilli ldquoAn op-timized path planning for the mobile robot using potential

field method and PSO algorithmrdquo Advances in IntelligentSystems and Computing vol 8 no 7 pp 139ndash150 2019

[4] R Li W Wu and H Qiao ldquoe compliance of robotichandsmdashfrom functionality to mechanismrdquo Assembly Auto-mation vol 35 no 3 pp 281ndash286 2015

[5] D C Robinson D A Sanders and EMazharsolook ldquoAmbientintelligence for optimal manufacturing and energy efficiencyrdquoAssembly Automation vol 35 no 3 pp 234ndash248 2015

[6] B Song Z Wang and L Sheng ldquoA new genetic algorithmapproach to smooth path planning for mobile robotsrdquo As-sembly Automation vol 36 no 2 pp 138ndash145 2016

[7] A Bakdi A Hentout H Boutami A Maoudj O Hachourand B Bouzouia ldquoOptimal path planning and execution formobile robots using genetic algorithm and adaptive fuzzy-logic controlrdquo Robotics and Autonomous Systems vol 89pp 95ndash109 2017

[8] Y Deng Y Liu and D Zhou ldquoAn improved genetic algo-rithm with initial population strategy for symmetric TSPrdquoMathematical Problems in Engineering vol 2015 Article ID212794 6 pages 2015

[9] B K Patle D R K Parhi A Jagadeesh and S K KashyapldquoMatrix-binary codes based genetic algorithm for pathplanning of mobile robotrdquo Computers amp Electrical Engi-neering vol 67 pp 708ndash728 2018

[10] M A Contreras-Cruz V Ayala-Ramirez and U H Hernandez-Belmonte ldquoMobile robot path planning using artificial beecolony and evolutionary programmingrdquo Applied Soft Comput-ing vol 30 pp 319ndash328 2015

[11] D Whitley D Hains and A Howe ldquoA hybrid genetic al-gorithm for the traveling salesman problem using generalizedpartition crossoverrdquo in Proceedings of the InternationalConference on Parallel Problem Solving from Naturepp 566ndash575 Springer Krakow Poland September 2010

[12] M Albayrak and N Allahverdi ldquoDevelopment a new mu-tation operator to solve the traveling salesman problem by aidof genetic algorithmsrdquo Expert Systems with Applicationsvol 38 no 3 pp 1313ndash1320 2011

[13] G Dantzig R Fulkerson and S Johnson ldquoSolution of a large-scale traveling-salesman problemrdquo Journal of the OperationsResearch Society of America vol 2 no 4 pp 393ndash410 1954

[14] M Nazarahari E Khanmirza and S Doostie ldquoMulti-ob-jective multi-robot path planning in continuous environmentusing an enhanced genetic algorithmrdquo Expert Systems withApplications vol 115 pp 106ndash120 2019

[15] J Ni K Wang H Huang et al ldquoRobot path planning basedon an improved genetic algorithm with variable lengthchromosomerdquo in Proceedings of the 2016 12th InternationalConference on Natural Computation Fuzzy Systems andKnowledge Discovery (ICNC-FSKD) pp 145ndash149 ChangshaChina August 2016

[16] C-C Tsai H-C Huang and C-K Chan ldquoParallel elite ge-netic algorithm and its application to global path planning forautonomous robot navigationrdquo IEEE Transactions on In-dustrial Electronics vol 58 no 10 pp 4813ndash4821 2011

[17] M M Malakiyeh S Shojaee and S Hamzehei JavaranldquoInsight into an implicit time integration method based onBezier curve and third-order Bernstein basis function forstructural dynamicsrdquo Asian Journal of Civil Engineeringvol 19 no 1 pp 1ndash11 2018

[18] N Arana-Daniel A A Gallegos C Lopez-Franco et alldquoSmooth global and local path planning for mobile robotusing particle swarm optimization radial basis functionssplines and Bezier curvesrdquo in Proceedings of the 2014 IEEE

Computational Intelligence and Neuroscience 9

Congress on Evolutionary Computation (CEC) pp 175ndash182Beijing China July 2014

[19] A A Neto DGMacharet andM FM Campos ldquoFeasible RRT-based path planning using seventh order Bezier curvesrdquo in Pro-ceedings of the IEEE International Conference on Intelligent Robotsand Systems pp 1445ndash1450 Taipei Taiwan October 2010

[20] B Song G Tian and F Zhou ldquoA comparison study on pathsmoothing algorithms for laser robot navigated mobile robotpath planning in intelligent spacerdquo Journal of Information andComputational Science vol 7 no 1 pp 2943ndash2950 2010

[21] K G Jolly R Sreerama Kumar and R Vijayakumar ldquoA Beziercurve based path planning in a multi-agent robot soccer systemwithout violating the acceleration limitsrdquo Robotics and Au-tonomous Systems vol 57 no 1 pp 23ndash33 2009

[22] Y J Ho and J S Liu ldquoCollision-free curvature-boundedsmooth path planning using composite Bezier curve based onVoronoi diagramrdquo in Proceedings of the 2009 IEEE Inter-national Symposium on Computational Intelligence in Roboticsand Automation-(CIRA) pp 463ndash468 Daejeon South KoreaDecember 2009

[23] M Elhoseny A arwat and A E Hassanien ldquoBezier curvebased path planning in a dynamic field using modified geneticalgorithmrdquo Journal of Computational Science vol 25pp 339ndash350 2018

[24] A arwat M Elhoseny A E Hassanien et al ldquoIntelligentBezier curve-based path planningmodel using chaotic particleswarm optimization algorithmrdquo Cluster Computing vol 22no S2 pp 4745ndash4766 2019

[25] L Han H Yashiro H T N Nejad et al ldquoBezier curve basedpath planning for autonomous vehicle in urban environ-mentrdquo in Proceedings of the 2010 IEEE Intelligent VehiclesSymposium pp 1036ndash1042 San Diego CA USA June 2010

[26] I Skrjanc and G Klancar ldquoOptimal cooperative collisionavoidance betweenmultiple robots based on BernsteinndashBeziercurvesrdquo Robotics and Autonomous Systems vol 58 no 1pp 1ndash9 2010

[27] K Yang and S Sukkarieh ldquoAn analytical continuous-cur-vature path-smoothing algorithmrdquo IEEE Transactions onRobotics vol 26 no 3 pp 561ndash568 2010

[28] S H Chia K L Su J H Guo et al ldquoAnt colony system basedmobile robot path planningrdquo in Proceedings of the 2010 FourthInternational Conference on Genetic and Evolutionary Com-puting pp 210ndash213 Shenzhen China December 2010

[29] A T Ismail A Sheta and M Al-Weshah ldquoA mobile robotpath planning using genetic algorithm in static environmentrdquoJournal of Computer Science vol 4 no 4 pp 341ndash344 2008

[30] I Chaari A Koubaa H Bennaceur et al ldquoSmartPATH ahybrid ACO-GA algorithm for robot path planningrdquo inProceedings of the 2012 IEEE Congress on EvolutionaryComputation pp 1ndash8 Brisbane QLD Australia June 2012

[31] K Liang Z J Chen and X Q Yan ldquoMobile robot pathplanning simulation researchrdquoModern Electronic Technologyvol 41 no 17 pp 167ndash172 2018

[32] MMMalakiyeh S Shojaee and S H Javaran ldquoDevelopmentof a direct time integration method based on Bezier curve and5th-order Bernstein basis functionrdquo Computers amp Structuresvol 194 pp 15ndash31 2018

[33] N Wang H Guo B Chen et al ldquoDesign of a rotary dielectricelastomer actuator using a topology optimization methodbased on pairs of curvesrdquo Smart Materials and Structuresvol 27 no 5 Article ID 055011 2018

[34] B Song Z Wang L Zou L Xu and F E Alsaadi ldquoA newapproach to smooth global path planning of mobile robots

with kinematic constraintsrdquo International Journal of MachineLearning and Cybernetics vol 10 no 1 pp 107ndash119 2019

[35] B Song Z Wang L Zou et al ldquoOn global smooth pathplanning for mobile robots using a novel multimodal delayedPSO algorithmrdquo Cognitive Computation vol 9 no 1pp 5ndash17 2017

10 Computational Intelligence and Neuroscience

sacrificing the path length e planned path length of theBCA-Q is 318503 which is 496 greater than that of theBCA It is noteworthy that the path length of BCA-Q is stillshorter than those of ACO and the GA Moreover itssmoothness and path security are greatly improved

5 Conclusions

In this paper a new smooth path planning method com-bining genetic operators and Bezier curves has been pro-posed for mobile robots First the control points of theBezier curve were determined by a genetic operation and thesmoothing characteristics of the Bezier curve were used tomake the planning path smoother and more consistentwhich reduced the energy loss in robot movement Secondthe safety distance was added to the fitness function andcould be dynamically adjusted according to the distancebetween the path and the obstacle to ensure the safe andefficient movement of the robot Finally the simulationresults showed that the proposed algorithm was effective infinding an optimal path this optimal path was shortersmoother and safer than those obtained by traditional al-gorithms However the simulation time was increased iswas because the improved algorithm paid more attention tothe quality of the solution in the process of path planningere still exist various meaningful topics to be addressedsuch as how to select the most appropriate number ofcontrol points of the continuous Bezier curve how to applythe algorithm to more complex practical environments(such as rooms or other special environments) and how toimprove the calculation efficiency with more control points

Data Availability

e data used to support the findings of this study are in-cluded within the article All required models and param-eters are listed in the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported in part by the National Key Re-search and Development Program of China under Grant2016YFE0104600 and the Science and Technology Programof Henan Province of China under Grant 172102410071

References

[1] M A P Garcia O Montiel O Castillo R Sepulveda andPMelin ldquoPath planning for autonomousmobile robot navigationwith ant colony optimization and fuzzy cost function evaluationrdquoApplied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[2] G Li andW Chou ldquoPath planning for mobile robot using self-adaptive learning particle swarm optimizationrdquo Science ChinaInformation Sciences vol 61 no 5 Article ID 052204 2018

[3] R K Mandava S Bondada and P R Vundavilli ldquoAn op-timized path planning for the mobile robot using potential

field method and PSO algorithmrdquo Advances in IntelligentSystems and Computing vol 8 no 7 pp 139ndash150 2019

[4] R Li W Wu and H Qiao ldquoe compliance of robotichandsmdashfrom functionality to mechanismrdquo Assembly Auto-mation vol 35 no 3 pp 281ndash286 2015

[5] D C Robinson D A Sanders and EMazharsolook ldquoAmbientintelligence for optimal manufacturing and energy efficiencyrdquoAssembly Automation vol 35 no 3 pp 234ndash248 2015

[6] B Song Z Wang and L Sheng ldquoA new genetic algorithmapproach to smooth path planning for mobile robotsrdquo As-sembly Automation vol 36 no 2 pp 138ndash145 2016

[7] A Bakdi A Hentout H Boutami A Maoudj O Hachourand B Bouzouia ldquoOptimal path planning and execution formobile robots using genetic algorithm and adaptive fuzzy-logic controlrdquo Robotics and Autonomous Systems vol 89pp 95ndash109 2017

[8] Y Deng Y Liu and D Zhou ldquoAn improved genetic algo-rithm with initial population strategy for symmetric TSPrdquoMathematical Problems in Engineering vol 2015 Article ID212794 6 pages 2015

[9] B K Patle D R K Parhi A Jagadeesh and S K KashyapldquoMatrix-binary codes based genetic algorithm for pathplanning of mobile robotrdquo Computers amp Electrical Engi-neering vol 67 pp 708ndash728 2018

[10] M A Contreras-Cruz V Ayala-Ramirez and U H Hernandez-Belmonte ldquoMobile robot path planning using artificial beecolony and evolutionary programmingrdquo Applied Soft Comput-ing vol 30 pp 319ndash328 2015

[11] D Whitley D Hains and A Howe ldquoA hybrid genetic al-gorithm for the traveling salesman problem using generalizedpartition crossoverrdquo in Proceedings of the InternationalConference on Parallel Problem Solving from Naturepp 566ndash575 Springer Krakow Poland September 2010

[12] M Albayrak and N Allahverdi ldquoDevelopment a new mu-tation operator to solve the traveling salesman problem by aidof genetic algorithmsrdquo Expert Systems with Applicationsvol 38 no 3 pp 1313ndash1320 2011

[13] G Dantzig R Fulkerson and S Johnson ldquoSolution of a large-scale traveling-salesman problemrdquo Journal of the OperationsResearch Society of America vol 2 no 4 pp 393ndash410 1954

[14] M Nazarahari E Khanmirza and S Doostie ldquoMulti-ob-jective multi-robot path planning in continuous environmentusing an enhanced genetic algorithmrdquo Expert Systems withApplications vol 115 pp 106ndash120 2019

[15] J Ni K Wang H Huang et al ldquoRobot path planning basedon an improved genetic algorithm with variable lengthchromosomerdquo in Proceedings of the 2016 12th InternationalConference on Natural Computation Fuzzy Systems andKnowledge Discovery (ICNC-FSKD) pp 145ndash149 ChangshaChina August 2016

[16] C-C Tsai H-C Huang and C-K Chan ldquoParallel elite ge-netic algorithm and its application to global path planning forautonomous robot navigationrdquo IEEE Transactions on In-dustrial Electronics vol 58 no 10 pp 4813ndash4821 2011

[17] M M Malakiyeh S Shojaee and S Hamzehei JavaranldquoInsight into an implicit time integration method based onBezier curve and third-order Bernstein basis function forstructural dynamicsrdquo Asian Journal of Civil Engineeringvol 19 no 1 pp 1ndash11 2018

[18] N Arana-Daniel A A Gallegos C Lopez-Franco et alldquoSmooth global and local path planning for mobile robotusing particle swarm optimization radial basis functionssplines and Bezier curvesrdquo in Proceedings of the 2014 IEEE

Computational Intelligence and Neuroscience 9

Congress on Evolutionary Computation (CEC) pp 175ndash182Beijing China July 2014

[19] A A Neto DGMacharet andM FM Campos ldquoFeasible RRT-based path planning using seventh order Bezier curvesrdquo in Pro-ceedings of the IEEE International Conference on Intelligent Robotsand Systems pp 1445ndash1450 Taipei Taiwan October 2010

[20] B Song G Tian and F Zhou ldquoA comparison study on pathsmoothing algorithms for laser robot navigated mobile robotpath planning in intelligent spacerdquo Journal of Information andComputational Science vol 7 no 1 pp 2943ndash2950 2010

[21] K G Jolly R Sreerama Kumar and R Vijayakumar ldquoA Beziercurve based path planning in a multi-agent robot soccer systemwithout violating the acceleration limitsrdquo Robotics and Au-tonomous Systems vol 57 no 1 pp 23ndash33 2009

[22] Y J Ho and J S Liu ldquoCollision-free curvature-boundedsmooth path planning using composite Bezier curve based onVoronoi diagramrdquo in Proceedings of the 2009 IEEE Inter-national Symposium on Computational Intelligence in Roboticsand Automation-(CIRA) pp 463ndash468 Daejeon South KoreaDecember 2009

[23] M Elhoseny A arwat and A E Hassanien ldquoBezier curvebased path planning in a dynamic field using modified geneticalgorithmrdquo Journal of Computational Science vol 25pp 339ndash350 2018

[24] A arwat M Elhoseny A E Hassanien et al ldquoIntelligentBezier curve-based path planningmodel using chaotic particleswarm optimization algorithmrdquo Cluster Computing vol 22no S2 pp 4745ndash4766 2019

[25] L Han H Yashiro H T N Nejad et al ldquoBezier curve basedpath planning for autonomous vehicle in urban environ-mentrdquo in Proceedings of the 2010 IEEE Intelligent VehiclesSymposium pp 1036ndash1042 San Diego CA USA June 2010

[26] I Skrjanc and G Klancar ldquoOptimal cooperative collisionavoidance betweenmultiple robots based on BernsteinndashBeziercurvesrdquo Robotics and Autonomous Systems vol 58 no 1pp 1ndash9 2010

[27] K Yang and S Sukkarieh ldquoAn analytical continuous-cur-vature path-smoothing algorithmrdquo IEEE Transactions onRobotics vol 26 no 3 pp 561ndash568 2010

[28] S H Chia K L Su J H Guo et al ldquoAnt colony system basedmobile robot path planningrdquo in Proceedings of the 2010 FourthInternational Conference on Genetic and Evolutionary Com-puting pp 210ndash213 Shenzhen China December 2010

[29] A T Ismail A Sheta and M Al-Weshah ldquoA mobile robotpath planning using genetic algorithm in static environmentrdquoJournal of Computer Science vol 4 no 4 pp 341ndash344 2008

[30] I Chaari A Koubaa H Bennaceur et al ldquoSmartPATH ahybrid ACO-GA algorithm for robot path planningrdquo inProceedings of the 2012 IEEE Congress on EvolutionaryComputation pp 1ndash8 Brisbane QLD Australia June 2012

[31] K Liang Z J Chen and X Q Yan ldquoMobile robot pathplanning simulation researchrdquoModern Electronic Technologyvol 41 no 17 pp 167ndash172 2018

[32] MMMalakiyeh S Shojaee and S H Javaran ldquoDevelopmentof a direct time integration method based on Bezier curve and5th-order Bernstein basis functionrdquo Computers amp Structuresvol 194 pp 15ndash31 2018

[33] N Wang H Guo B Chen et al ldquoDesign of a rotary dielectricelastomer actuator using a topology optimization methodbased on pairs of curvesrdquo Smart Materials and Structuresvol 27 no 5 Article ID 055011 2018

[34] B Song Z Wang L Zou L Xu and F E Alsaadi ldquoA newapproach to smooth global path planning of mobile robots

with kinematic constraintsrdquo International Journal of MachineLearning and Cybernetics vol 10 no 1 pp 107ndash119 2019

[35] B Song Z Wang L Zou et al ldquoOn global smooth pathplanning for mobile robots using a novel multimodal delayedPSO algorithmrdquo Cognitive Computation vol 9 no 1pp 5ndash17 2017

10 Computational Intelligence and Neuroscience

Congress on Evolutionary Computation (CEC) pp 175ndash182Beijing China July 2014

[19] A A Neto DGMacharet andM FM Campos ldquoFeasible RRT-based path planning using seventh order Bezier curvesrdquo in Pro-ceedings of the IEEE International Conference on Intelligent Robotsand Systems pp 1445ndash1450 Taipei Taiwan October 2010

[20] B Song G Tian and F Zhou ldquoA comparison study on pathsmoothing algorithms for laser robot navigated mobile robotpath planning in intelligent spacerdquo Journal of Information andComputational Science vol 7 no 1 pp 2943ndash2950 2010

[21] K G Jolly R Sreerama Kumar and R Vijayakumar ldquoA Beziercurve based path planning in a multi-agent robot soccer systemwithout violating the acceleration limitsrdquo Robotics and Au-tonomous Systems vol 57 no 1 pp 23ndash33 2009

[22] Y J Ho and J S Liu ldquoCollision-free curvature-boundedsmooth path planning using composite Bezier curve based onVoronoi diagramrdquo in Proceedings of the 2009 IEEE Inter-national Symposium on Computational Intelligence in Roboticsand Automation-(CIRA) pp 463ndash468 Daejeon South KoreaDecember 2009

[23] M Elhoseny A arwat and A E Hassanien ldquoBezier curvebased path planning in a dynamic field using modified geneticalgorithmrdquo Journal of Computational Science vol 25pp 339ndash350 2018

[24] A arwat M Elhoseny A E Hassanien et al ldquoIntelligentBezier curve-based path planningmodel using chaotic particleswarm optimization algorithmrdquo Cluster Computing vol 22no S2 pp 4745ndash4766 2019

[25] L Han H Yashiro H T N Nejad et al ldquoBezier curve basedpath planning for autonomous vehicle in urban environ-mentrdquo in Proceedings of the 2010 IEEE Intelligent VehiclesSymposium pp 1036ndash1042 San Diego CA USA June 2010

[26] I Skrjanc and G Klancar ldquoOptimal cooperative collisionavoidance betweenmultiple robots based on BernsteinndashBeziercurvesrdquo Robotics and Autonomous Systems vol 58 no 1pp 1ndash9 2010

[27] K Yang and S Sukkarieh ldquoAn analytical continuous-cur-vature path-smoothing algorithmrdquo IEEE Transactions onRobotics vol 26 no 3 pp 561ndash568 2010

[28] S H Chia K L Su J H Guo et al ldquoAnt colony system basedmobile robot path planningrdquo in Proceedings of the 2010 FourthInternational Conference on Genetic and Evolutionary Com-puting pp 210ndash213 Shenzhen China December 2010

[29] A T Ismail A Sheta and M Al-Weshah ldquoA mobile robotpath planning using genetic algorithm in static environmentrdquoJournal of Computer Science vol 4 no 4 pp 341ndash344 2008

[30] I Chaari A Koubaa H Bennaceur et al ldquoSmartPATH ahybrid ACO-GA algorithm for robot path planningrdquo inProceedings of the 2012 IEEE Congress on EvolutionaryComputation pp 1ndash8 Brisbane QLD Australia June 2012

[31] K Liang Z J Chen and X Q Yan ldquoMobile robot pathplanning simulation researchrdquoModern Electronic Technologyvol 41 no 17 pp 167ndash172 2018

[32] MMMalakiyeh S Shojaee and S H Javaran ldquoDevelopmentof a direct time integration method based on Bezier curve and5th-order Bernstein basis functionrdquo Computers amp Structuresvol 194 pp 15ndash31 2018

[33] N Wang H Guo B Chen et al ldquoDesign of a rotary dielectricelastomer actuator using a topology optimization methodbased on pairs of curvesrdquo Smart Materials and Structuresvol 27 no 5 Article ID 055011 2018

[34] B Song Z Wang L Zou L Xu and F E Alsaadi ldquoA newapproach to smooth global path planning of mobile robots

with kinematic constraintsrdquo International Journal of MachineLearning and Cybernetics vol 10 no 1 pp 107ndash119 2019

[35] B Song Z Wang L Zou et al ldquoOn global smooth pathplanning for mobile robots using a novel multimodal delayedPSO algorithmrdquo Cognitive Computation vol 9 no 1pp 5ndash17 2017

10 Computational Intelligence and Neuroscience