stochastic c-gnet environment modeling and path planning...

Research ArticleStochastic C-GNet Environment Modeling and Path PlanningOptimization in a Narrow and Long Space

Jianjian Yang Zhiwei Tang XiaolinWang Zirui Wang Biaojun Yin andMiaoWu

School of Mechanical Electronic and Information Engineering China University of Mining and Technology Beijing 100083 China

Correspondence should be addressed to Jianjian Yang yangjiannedved163com

Received 5 January 2018 Revised 26 April 2018 Accepted 16 May 2018 Published 19 June 2018

Academic Editor Salvatore Alfonzetti

Copyright copy 2018 Jianjian Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This study proposes a novel method of optimal path planning in stochastic constraint network scenarios We present a dynamicstochastic grid network model containing semienclosed narrow and long constraint information according to the unstructuredenvironment of an underground or mine tunnel This novel environment modeling (stochastic constraint grid network) computesthe most likely global path in terms of a defined minimum traffic cost for a roadheader in such unstructured environmentsDesigning high-dimensional constraint vector and traffic cost in nodes and arcs based on two- and three-dimensional terrainelevation data in a grid network this study considers the walking and space constraints of a roadheader to construct the networktopology for the traffic cost value weights The improved algorithm of variation self-adapting particle swarm optimization isproposed to optimize the regional path The experimental results both in the simulation and in the actual test model settingsillustrate the performance of the described approach where a hybrid centralized-distributedmodelingmethod with path planningcapabilities is used

1 Introduction

Environment modeling refers to the use of a unified formof expression to complete the environmental entitys ownattributes and the internal structure of the entity modelnot only to describe the static attributes but also to expressdynamic changes The study on environment modeling forpath planning at home and abroad mainly focuses on amobile robot [1ndash4] mostly in an indoor or outdoor structuralenvironment such as an unmanned vehicle (eg rescuevehicles exploration car lunar rover and Mars rover) Tra-ditional environmental modeling methods mainly includeV-Graph [5] T-Graph [6] and Voronoi [7] (eg spacemethod artificial potential fieldmethod [8] and approximateelement decomposition method [9] such as topology graphand mathematical analysis class) These methods usuallyuse external measurement to obtain the path planning ofterrain two-dimensional coordinates and three-dimensionalaltitude information to establish digital terrain [10ndash13] andestimate its trafficability [14] These traditional approachessolve the path planning problem to achieve two goals shortest

path [15ndash17] and obstacle avoidance demand [18] Howeverin the face of an unstructured dynamic path environmentformed by geological structure and stochastic dynamic gen-eration traditional space environment modeling should alsoconsider topography and geology pose adjustment cost andwalking constraints in addition to the realization of theshortest path planning target and obstacle avoidance

Shi Meiping et al [19] proposed a lunar surface multi-constraint modeling method while Chen Cheng et al [20]introduced a kind of a long-range polar robot environmentmodeling method Pablo Urcola [21] et al proposed a pathplanning method of a multirobot shortest path randomterrain Li Tiancheng et al used a fan-shaped grid networkto perform research robot path planning in a static envi-ronment Traditionally the existing network path model isestablished using the orthogonal grid In these methods thedigital terrain is determined by considering the factors ofmotion constraint system uncertainty terrain availabilityand so on However this method is generally applicableto the static structural environment model Most of theroadway engineering paths are dynamically generated and

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 9452708 13 pageshttpsdoiorg10115520189452708

2 Mathematical Problems in Engineering

Figure 1 Descartes coordinate grid map

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Figure 2 Polar coordinate grid map

the floor surface distribution is stochastic The particularityof the walking constraint and the pose adjustment costof the underground roadheader are not reflected in theabove-mentionedmethodsThereforemost of these dynamicstochastic unstructured environment modeling problemscan be more generally classified as a stochastic constraintnetwork

Based on the above this study investigates the stochasticnetwork unstructured spatial environment modeling andavailability problem and increases the high-dimensional con-straint vector and traffic cost in the constraint grid network(C-GNet) nodes and arcs based on grid two-dimensional(2D) terrain data and three-dimensional (3D) altitude dataA network topology with the walking cost weights is con-structed considering the space and the walking constraintsof the roadheader The minimum cost of the network nodesand arcs was determined as the objective function to proposethe variation self-adapting particle swarm optimization algo-rithm and optimize the region path

2 Materials and Methods

21 Environment Model The classical grid network partitionenvironment model mainly includes a Descartes coordinategrid model (Figure 1) and a polar coordinate grid model(Figure 2) which represents the two-dimensional positioninformation119873119890119905 = (119875119870 119865) is a conventional grid network P is aset of nodes representing the grid position point (119909119894 119910119894) K isthe set of altitudes representing the altitude value of the grid

Figure 3 C-GNet diagram structure

position point and F is the set of arcs representing the nodeconnection path

The classical grid network 2D terrain data and the 3Daltitude data cannot reflect other dimensional data such astraffic cost and geological influence cost in the modelingof some stochastic dynamic network environment modelswithout considering the numerical representation of randomdistribution and the behavior constraints of the roadheaderTherefore performing research on the multidimensionalconstraint grid network and traffic cost information is ofpractical significance

Definition 1 The 6-tuple sum = (119875119870 119865119882119883119872) is a gridnetwork system which is the constraint grid network (C-GNet) according to the following structural rules

(1) 119873119890119905 = (119875119870 119865) is a conventional grid network(2) 119870 119875 rarr 119911119894 = 119891(119909119894 119910119894) is the altitude function of

C-GNet(3) W 119865 rarr 119908119894 = 119891(119909119894 119910119894 119911119894 119905119892) is the weight function

of C-GNet(4) X 119865 rarr 119883119894 119894 = 1 2 3 4 is the constraint variant set

of C-GNet(5) MPrarrZ (set of nonnegative integer) is the token or

state

The surface traffic attribute information of the con-strained grid node was established (Figure 3) The proper-ties of the C-GNet structure parameters were defined asProperty 2

Property 2 sum = (119875119870 119865119882119883119872) which satisfies the fol-lowing conditions(1) 119901 isin 119875 is any node ofC-GNet ifexist1199011 1199012 119901119873makingforall1 le 119894 le 119873 Δ119901119894 = 119901 | (119909119894minus1 119910119894minus1)119880(119909119894 119910119894minus1)119880(119909119894+1119910119894minus1)119880(119909119894minus1 119910119894+2)119880(119909119894 119910119894+2) U(119909119894+1 119910119894+2) sub 119875 and Δ119901 = 119901ΔΔ119901 and 119901Δ are the preset and postset of 119901(2) 119891 isin 119865 119891119894119894+1 = 119891(119901119894119894 rarr 119901119894119894+1) represents thedirected arc between (119909119894 119910119894) and (119909119894 119910119894+1)The arc orientationdepends on the starting position of the node120596 represents thearc length weight of C-GNet 120579 represents the arc angle con-straint The state transition (position transfer) between theadjacent nodes is called transition 120595119894minus119895 119872(119901119894) rarr 119872(119901119895)(3) In the C-GNet the path from the source node s tothe target node d cannot be closed

997888997888997888rarr119903119900119886119889 119901119904 120595119904minus119886 119901119886 120595119886minus119887 119901119889997888997888997888rarr1199031199001198861198891(119904 119889)997888997888997888rarr1199031199001198861198892(119904 119889) 997888997888997888rarr119903119900119886119889119899(119904 119889) represent the

Mathematical Problems in Engineering 3

road from the source nodes to the target nodes 119877(119904 119889)represents the set of paths and the total cost expressionfor a path is 119882119903(119904 119889) = sum119889119894=119904 119908119894 = sum119889119894=119904 119891 119894119905119903119886V119890119897minus119888119900119899119889119894119905119894119900119899 Themathematical expression of the optimal path 119877(119904 119889) is trans-formed into solving the minimum cost path in the path set

119882min (119904 119889) = min( 119889sum119894=119904119895

119908119894120595119894minus119895)

= min( 119889sum119894=119904119895

119891119894119905119903119886V119890119897-119888119900119899119889119894119905119894119900119899120595119894minus119895)(1)

119889sum119894=119904119895

120595119894minus119895 le 10038161003816100381610038161198751199031198991199061198981003816100381610038161003816 minus 1 (2)

2 le 10038161003816100381610038161198751199031198991199061198981003816100381610038161003816 le 119898 + 1 (3)

119875119903119899119906119898 sub 2 3 4 119898 + 1 (4)

120595119894minus119895 = 0 997888997888997888rarr119903119900119886119889 (119894 119895) notin 119877 (119904 119889)1 997888997888997888rarr119903119900119886119889 (119894 119895) isin 119877 (119904 119889) (5)

In the traffic cost model 119882min(119904 119889) is the objective functionvalue Equations (2) to (5) denote the constraint functionThe transition120595119894minus119895 indicates whether the destination path hasarcs119891119894minus119895 = 119891(119901119894 rarr 119901119895) 119875119903119899119906119898 is the subset of all nodes inthe target path |119875119903119899119906119898| is the number of all nodes in the set120595119894minus119895 = 0 indicates that the node transition does not belong tothe target path 120595119894minus119895 = 1 indicates the opposite situation

The traffic cost of the path network model is establishedthe nodes and the arcs of which have stochastic attributesdetermined as Definition 3

Definition 3 The network stochastic characteristics aremainly embodied in the weight function W of the C-GNetarc 119865 rarr 119891119894119905119903119886V119890119897minus119888119900119899119889119894119905119894119900119899=119891(119909119894 119910119894 119911119894119905119892)

119894=1119873119892=123 When the arc weight

function of the network node dynamic randomly changes119882(119894) which is the arc weight function becomes a stochasticprocess Equation (6) denotes the dynamic stochastic con-straint grid net (C-GNet)

sumlowast = (119875119870 119865119882 (119894) 119883119872) (6)

The cost function 119882(119894) 119894 isin 119868 in the dynamic stochasticC-GNet obeys the discrete stochastic process distribution119868 is the discrete sequence number and 119882(119894) represents thediscrete distribution In an instance of a specific networknode sequence number I 119882(119894) is a discrete stochastic vari-able whose corresponding probability is expressed as 119901(119894) =119901(120595119894minus119895) and sum119894 119901(119894) = sum119894+1119895+1119894=119894minus1119895=119895minus1 119901(120595119894minus119895) = 122 Guidance Factor and Network Cost Model of C-GNetThe C-GNet model has the path guidance characteristicsPath planning mainly combines the geological and guidance

factors of the special environment The guidance factorformula is defined as follows

119892 (119894) = |119889 (119894)| times 119886 (7)

where 119894 represents the order number of the path points to beselected 119894 = 2 3 9 119889 is the distance between the centralline of each path point and 119886 is the guide coefficient and spanbelonging to [0 1] Its size reflects the influence of the pathpoint deviation from the degree of center to the overall path

We propose herein the concept of centripetal degreebased on each path point corresponding to a guidance factor119892 (see (8)) Consider

119866 = 119873sum119894=1

119892 (119894) (8)

The information contained in the arc is (119868119899119891 120579119903 120579119901 120579119905 119871)where 119868119899119891 indicates the geological cost 120579119903 indicates therolling angle of the arc ground surface 120579119901 indicates the angleof pitch 120579119905 indicates the steering angle and119871 is the arc length

119882119903 = 119873minus1sum119894=1

(119868119899119891119894 + 120579119903119894 + 120579119901119894 + 120579119905119894 + 119871 119894) (9)

In (9) 119894 = 1 2 3 119873 minus 1 where N represents the numberof nodes for path planningThe total cost formula of the pathis denoted in

119882119903 (119904 119889) = 119882 + 119866 (10)

Therefore the optimal path planning aims to find thecomprehensive cost of min(119882119903(119904 119889))23 Variation Self-Adapting Particle Swarm Optimization(VSAPSO) This study compared and verified four kindsof PSO algorithms including this method to verify theVSAPSO algorithm performance The simulation experi-ments of this method were completed under MATLABR2016a The operation system was Windows 10 64-bit opera-tion system while the CPU was i5-4590

231 Algorithm Principle The classical particle swarm opti-mization algorithm [22] is expressed as

V119894119889 (119905 + 1) = 119908 times V119894119889 (119905) + 1198881 times 1199031 times (119901119894119889 minus 119909119894119889 (119905)) + 1198882times 1199032 times (119901119892119889 minus 119909119894119889 (119905)) (11)

119909119894119889 (119905 + 1) = 119909119894119889 (119905) + V119894119889 (119905 + 1) (12)

V119894119889 =

Vmax V119894119889 gt Vmax

minusVmax V119894119889 lt minusVmax

V1198941198891003816100381610038161003816V1198941198891003816100381610038161003816 lt Vmax

(13)

119908 (119905) = 119908max minus 119908max minus 119908min119879max119905 (14)


where V119894119889 is the speed of the current particle 119894 119909119894119889 is theposition of the current particle 119894 1198881 and 1198882 are the learningfactors 1199031 and 1199032 are random numbers among [0 1] 119901119894119889 isthe personal best position of the current particle 119894 119901119892119889 isthe global best position of the current particle 119894 Vmax is themaximum speed 119908119898119886119909 is the maximum inertia weight 119908119898119894119899is the minimum inertia weight t is the number of iterationsand 119879119898119886119909 is the maximum number of iterations

This method has the advantages of a simple principlea few controlled parameters and a fast convergence speedHowever it also has the disadvantage of easy convergenceto local optimal and low search precision Therefore theVSAPSO algorithm was proposed in this study

119908 = 119908max minus (119908max minus 119908min) times (119891max minus 119891119894119905119899119890119904119904 (i))119891max minus 119891min(15)

where 119891119894119905119899119890119904119904(119894) indicates the fitness for each generation of119894 particles 119891max indicates the maximum fitness and 119891minindicates the minimum fitness

This proposed algorithm can automatically adjust theparameters of the inertia weight w according to the currentparticle fitness The w value increases when the fitness valueis large thereby improving the search speed and the globalsearch ability of particlesThew value becomes smaller whenthe fitness value is small thereby reducing the particle searchspeed and improving the local search ability of particles

The modified inertia weight w has a larger adjustmentrange which improves the search capability of the algorithmWe used herein the way of mutation in the genetic algorithmto enable the particles to randomly reset positions at acertain probability such that they can jump out of theoriginal location to research thereby reducing the probabilityof particle swarm falling into local minimum The way ofmutation is shown in (16) as follows

119909119894119889 = 119909max times 119903119886119899119889119904 (1 119863) 119903119886119899119889 () ge 119901119909119894119889 119903119886119899119889 () lt 119901 (16)

where p represents a mutation threshold 119909max represents themaximum value allowed by the location and rands (1 D)represents a D random number between 0 and 1

232 VSAPSO Performance Test We used herein four kindsof test functions to test the standard PSO algorithm adaptivePSOalgorithmwithmodified inertiaweight formula (APSO)adaptive weight particle swarm optimization algorithm withthe compression factor (CFWPSO) and VSAPSO

The four kinds of PSO algorithms have the same basicparameters particle swarm size 119873 = 40 maximum speed119881max = 1 maximum inertia weight 119908max = 09 minimuminertia weight119908min = 04 and learning factor 1198881 = 1198882 = 205where the VSAPSO mutation threshold is 119901 = 098 Theend condition of the four kinds of algorithms is to reach themaximum iteration step of 100 steps and the four kinds oftest functions are all four-dimensional functions

When the maximum number of iterations is achievedthe algorithm runs 10 times to obtain the fitness decreasing

PSOCFWPSO

APSOVSAPSO

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alue

10 20 30 40 50 60 70 80 90 1000Iterations number

Figure 4 Fitness curve decreasing under the Griewank function

PSOCFWPSO

APSOVSAPSO

times106

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alue


Figure 5 Fitness curve decreasing under the Rosenbrock function

PSOCFWPSO

APSOVSAPSO

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alue


Figure 6 Fitness curve decreasing under the Rastrigin function

curve of each algorithmwhen it reaches the minimum fitnessfunction value under each test function (Figures 4ndash7)

The initial values of the algorithms were randomlyassigned hence the starting points of the curves weredifferent As shown in Figures 4ndash7 the PSO and APSOalgorithms illustrate that the curve does not decrease inthe number of iterative steps when in convergence Theeffect of the CFWPSO algorithm was consistent with thetwo above-mentioned items Under the Ackley functionshown in Figure 7 the fitness value of the PSOAPSO andCFWPSO algorithms even occurred over 20 in 100 steps


PSOCFWPSO

APSOVSAPSO


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Figure 7 Fitness curve decreasing under the Ackley function

Table 1 Minimum fitness

Test function PSO APSO CFWPSO VSAPSOGriewank 0327 07778 03195 2220eminus15Rosenbrock 0031 6697eminus4 00011 7459eminus18Rastrigin 355eminus15 628eminus7 188eminus10 0Ackley 19777 16717 19837 7808eminus7

Table 2 Variance

Test function PSO APSO CFWPSO VSAPSOGriewank 0552 0146 02653 1261eminus22Rosenbrock 1578 583 70813 5825eminus27Rastrigin 2605eminus21 2453eminus9 55779eminus13 0Ackley 1319 0037 00186 2876eminus9

because of the lack of global search ability and local minimawhen these three algorithms were convergent leading to theinability to converge to the minimum To sum up underthe four functions the VSAPSO algorithm had a strongerglobal search ability and was better than the PSOAPSO andCFWPSO algorithms in the decline speed

Thefitness function is the optimization objective functionof the PSO algorithm In this paper the comprehensive costfunction of path planning is our fitness function Tominimizethe energy and time consumption of the roadheader theminimum value of the comprehensive cost function is setas the optimization target of the PSO algorithm The PSOalgorithm itself has the randomness of optimization There-fore there will be a certain fluctuation in the optimizationresults of the algorithm It is necessary to evaluate theadvantages and disadvantages of the algorithm through themathematical statistics When the maximum number ofiterations is achieved the algorithm runs 10 times to obtainthe minimum fitness function value variance and mean ofeach algorithm under each test function (Tables 1ndash3)

The inertia weight adjustment strategy proposed in thispaper can adjust the search speed of particles according to theposition of particles And the convergence of the algorithmalso can be increased The minimum fitness value of thealgorithm can be used to evaluate the limit optimizationability of the algorithm

Table 3 Average value


Table 1 showed that the minimum fitness function valueof the VSAPSO algorithm proposed in this study was thehighest compared with that of the PSOAPSO and CFWPSOalgorithms The convergence accuracy was the highest

Variance of minimum fitness represents the fluctuationof minimum fitness function value obtained by algorithmoptimization The probability that different improved PSOalgorithm can stabilize a certain value is also differentThe smaller the fluctuation the better the stability of thealgorithm

Average of minimum fitness represents the optimizationability of the algorithm in the usual case and represents thenoncontingency of the minimum value that the algorithmcan reach If the minimum value of the algorithm can beachieved many times it shows that the algorithm can reachsmaller fitness value by chanceThis noncontingency is moreappropriate in the use of mean value in the mathematics

In Tables 2 and 3 it was presented that the average valueand the variance of the VSAPSO algorithm were smallerthan those in the PSOAPSO and CFWPSO algorithms andindicated that the stability and the convergence of VSAPSOwere the highest The inertia weight adjustment strategy andthe compilation strategy proposed in this paper made theVSAPSO algorithm more stable to find the minimum fitnessvalue If the minimum fitness value generally deviated fromthe mean value the stability and reliability of the algorithmwere not high On the contrary it showed that the algorithmhas high stability and credibility Consequently the proposedVSAPSO algorithm had a faster speed and a higher precisionin convergence than the other three algorithms

3 Results and Discussion

31 Environment Modeling Because of the special under-ground coal mine traveling environment in this paper itbelonged to the narrow space roadway terrain structureThere were the strong coal dust humid air weak light andthe threat of gas to human and the electrical and electronicequipment in coal mine These posed a great obstacle topath planning of roadheader and it was difficult to carryout field test in the coal mine before the verified theorymethodTherefore the authors and their teamhad carried outrelated test research on the position and orientation detectionmethod of roadheader in the ground environment such aslaser automatic measurement method

The simulation coal mine roadway used in this paper wasbased on the size and driving data of EBZ type roadheader Itwas a special lightweight roadheader which was suitable forcoal roadway half coal rock roadway and soft rock roadway


Figure 8 Simulation model of roadway

and tunnel heading The maximum cutting height was 28mand the maximum cutting bottom width was 352m Theeffective section area is 98m2 and wider cutting width wasless than 16 degrees

By using the laser automatic measurement method theposition information of the roadheader in the tunnel wasmeasured to determine the coordinate value in the mapand the position and posture information was measuredThe roll angle the pitch angle and the steering angle of theroadheader were obtained by the solution which providedthe actual value of the cost function value solution of themethod proposed in this paper The test basis and the stepsof the measurement were

(a)Model the laneway and deploy total stationmeasuringplatform rear view prism and prism to be measured

(b) Level and adjust the total station the tested prism andthe rear viewing prism

(c) Use the total station to collect the position andorientation data of the roadheader

(d) Calculate the position and orientation data of theroadheader in order to obtain the roll angle pitch anglesteering angle and location information of the roadheader

The experimental environment and real view of experi-mental scenario were shown in Figures 9sim11

The experimental data showed that the positioning accu-racy of the Y axis was better than that of 10cm and thepositioning accuracy of X and Z axis was better than that of2cm under the distance of 12m spacing and the distance of17m between the launching station and the roadheader

Through the data acquisition process of the position andposture of the above roadheader the information needed forthe calculation of the cost function value in the C-G gridmapcould be obtainedThe data obtained from themeasured datacould be brought into the roadheader cost function modelproposed in this paper Based on this test data stochastic C-G grid map proposed in this paper could be set up (Figure 8)and theVSAPSO algorithmwas used to excavate theC-G gridmap information to plan optimal path of roadheader

The establishment steps of the stochastic C-GNet envi-ronment model are as follows(1) The single path forward distance of the roadheaderis determined according to the step distance of the driftingmachine 119871 119894 = 08mThe transverse distance is 4 m(2) The node resolution of the constraint grid is deter-mined according to the stepping control precision of the

Figure 9 Laser automatic measurement

Figure 10 Position measurement device

Figure 11 Real view of experimental scenario

roadheader 01 m The parameter assignments of 119899 isin 119873 and119873 = 9 in 997888997888997888rarr119903119900119886119889119899(119904 119889) are determined(3) The geological cost 119891119894119905119903119886V119890119897minus119888119900119899119889119894119905119894119900119899 = (119868119899119891119894 + 120579119903119894 +120579119901119894 + 120579119905119894)120595119894minus119895 119895 = 119899119890119909119905(119894) is determined according to thesum of the total cost weights based on the node of the path119882119903(119904 119889) = sum119889119894=119904 119908119894 = sum119889119894=119904 119891119894119905119903119886V119890119897minus119888119900119899119889119894119905119894119900119899


215

105

0minus0508

minus10604

minus1502 0 minus2

002040608

1

005

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045

Figure 12 Cost simulation obeying the normal distribution

(4) According to the behavior constraints of the road-header in the narrow and long tunnel in Figure 9 formula119895 = 119899119890119909119905(119894) indicates the postposition node that is thelongitudinal direction of the current node with 45∘ 90∘ and135∘ as the three direction nodes(5) The precision position measurement system of theroadheader in the mine determines the 3D coordinates andcompletes the calculation of the pitch roll and deflectionThe reliability of themethod is verified by the simulation databased on the value of 119891119894119905119903119886V119890119897minus119888119900119899119889119894119905119894119900119899(6) From the actual roadheader tunneling environmentthe roadway is close to the middle line after tunneling Thecoal-receiving condition here is relatively good whereas thaton both sides of the roadways is relatively poorTherefore thecloser the roadway floor condition the better the distanceand the worse the middle line condition According to thedistance between the nodes and the middle line the costvalue distribution of the arc path is assumed to be a normaldistribution The grid network model is set up as shown inFigure 12 We first calculate the standard normal distributionof the interval [minus2 2] and then take 41 intervals of 01 normaldistribution values Each arc is determined according to thecontent of Definition 3 (see (17)) Consider

119882(119894) = 119891119894119905119903119886V119890119897minus119888119900119899119889119894119905119894119900119899 = 119903119886119899119889 (1) times (05 minus 119873 (119894))05 (17)

where 119903119886119899119889(119894) indicates the random number between 0 and 1119873(119894) represents the normal distribution value correspondingto the arc and 119894 is the number of network nodes in 119894 =1 2 41(7)We set the centrality of cost 119892(119894) = |119863119894| times 119886119892 accordingto the distanceD from the center of the roadheader pathD isthe distance of each path from the center line in centimetersthe value of which is for the dimensionless form of thedistance 119886119892 represents the guidance coefficient |119886119892| isin [0 1]The centripetal degree concentration indicates the influenceof the path point deviation from the center degree to theoverall path The greater the coefficient the closer the pathcloser to the middle line Herein 119886119892 = 05 and the numericalvalue aims to satisfy the path point at a feasible interval andguide the path to travel along the center line(8) Integrating steps 6 and 7 we propose herein the trafficcost formula119882119903(119904 119889) = 119882 + 119866 = sum119882(119894) + sum119892(119894) Figure 10

Table 4 Comparison of path-planning results

Type Minimum fitness Mean VariancePSO 1304 1372 0062CFWPSO 1154 1284 0009VSAPSO 1071 1176 0007

shows the network traffic cost model with the horizontaldirection for the 41 points and the longitudinal axis nine stepsof the roadheader An example of random starting nodes(0 0) was established as a grid map based on this (Figure 13)

There were three algorithms used to plan path of theroadheader Each path planning algorithm runs 10 times andthe path of theminimumfitness is obtained as planned by thethree algorithms shown in Figures 14 and 15

Figures 14 and 15 show that the coincidence path of the0sim01 part in the diagramwas planned by PSO andCFWPSOwhile 02sim03 part is determined by CFWPSO and VSAPSOThe coincidence path of the 04sim05 part in the diagramwas planned by PSO and VSAPSO while 05sim08 part isdetermined by CFWPSO and PSO

Theminimum comprehensive cost function value meanand variance of the statistical algorithmafter running 10 timeswere shown in Table 4

It could be seen from Table 4 that compared with theother two PSO algorithms the minimum comprehensivecost function of the VSAPSO algorithm was minimizedby comparing the minimum comprehensive cost functionvalue under the premise of the stochastic starting point ofthe path planning Thus the VSAPSO algorithm had betteroptimization ability in the path planning problem Accordingto the comparison of variance the result of path planningusing the VSAPSO algorithm was smaller which indicatedthat the path planning was more stable using the VSAPSOalgorithm Furthermore the mean value of the path planningusing the VSAPSO algorithm was smaller by comparing themean value It showed that the path planning was obtainedby using the VSAPSO algorithm The probability of smallercomposite cost function value was higher and the result wasmore believable

On the premise of stochastic path planning and stoppoint the VSAPSO algorithm was more efficient in thethree algorithms and the results were more stable and morereliable

32 Stochastic C-GNet Test under Multitarget Points Takingthe actual working conditions of the roadheader in the tunnelas an example the cutting boundary can be compensatedby the motion of the cutting arm hence the moving targetpoint cannot be unique In the simulation model we definethe set of stochastic C-GNet target nodes as 119863119903 119863119903(119894) isin[minus03 03] and the source node 119875119904 isin 119878119903 as the random sourcenode (02 0) The VSAPSO algorithm was used to simulatethe multitarget set random C-GNet test under the guidancefactor

Figure 16 shows that the path point 119875119889 was found underthe above-mentioned conditions of the grid map 119875119889 notin 119860 119903


Table 5 Minimum integrated cost value corresponding to each path

Target minus05 08 minus04 08 minus03 08 minus02 08 minus01 08 0 08 01 08 02 08 03 08 04 08 05 08Total cost 18 17 15 14 12 107 11 15 215 216 216

Path with Guidance factor

0

005

01

015

02

025

03

035

minus1

minus15 0

04

02

01

03

05

minus03

minus01

minus05

minus04

minus02 21

15

0

01

02

03

04

05

06

07

08

Figure 13 Stochastic C-GNet environment model

002040608

minus15 minus1 minus05 0 05 15minus2 21

PSO pathCFWPSO pathVSAPSO path

Accessible pointsCutting compensation allowable point

Figure 14 Path planning under simple C-Grid network

Therefore the guidance factor G was added based on the arcconsumption Figure 17 presents the planning path when theguidance factor proportion was large We can see from thefigure that the guiding factor can implement the rectificationwork path well However if we consider that the guidingfactor will cause a large roadheader traveling traffic cost wemust consider the compound factors of the guiding factorand the consumption values (Figure 18) after adjusting theguiding factor path planning algorithm

The adjustment strategy was made according to thecomprehensive influence factors of the geological traffic andthe guidance factors in the above-mentioned terrain Anexample of random starting nodes (02 0) was established asa traffic cost model (Figure 19) and as a grid map based onthis (Figure 20)

We performed path planning for all sets of target pointsusing the VSAPSO algorithm (Figure 21)

Table 5 shows the cost 119882(119894) of the simulated net-work nodes obeying a normal distribution target nodes119875119889 = (0 0) minimum traffic cost 119882119903(119904 119889) and best path(Figure 22)

After fully considering the stochastic characteristics ofthe floor and the surrounding rock surface of tunnelingafter roadheader cutting the cost function of the simulationroadway and the simulation modeling of the roadheaderspath planning were completed In addition the adaptabilityof the theoretical method was verified Under this theoreticalmodel the practical measurement and path planning need tofurther improve the reliability of the topographic survey dataand prior knowledge of geological conditions


minus15 minus1

PSOCFWPSOVSAPSO

0

01

02

03

04

05

06

07

08

21

150

03

04

02

05

01

minus02

minus01

minus05

minus04

minus03

0

005

01

015

02

025

03

035

04

045

Figure 15 Comparison path planning under stochastic C-GNet environment model

single target path

minus1

minus15 0

04

02

01

03

05

minus03

minus01

minus05

minus04

minus02 21

15

0

01

02

03

04

05

06

07

08

0

005

01

015

02

025

03

035

04

045

Figure 16 Single target path

4 Conclusions

This work studied the stochastic network optimal path plan-ning problem and analyzed the network structure character-istics stochastic nature and cost model of network arcs and

nodes We proposed a simulation for the stochastic processby analyzing the embodiment of the behavioral constraintsin the network model We took the 2D grid expansionfor the high-dimensional grid computing cost model andcomprehensive cost value into a 3D grid lower dimensional



0

01

02

03

04

05

06

07

08

minus1

minus15 0

04

02

01

03

05

minus03

minus01

minus05

minus04

minus02 21

15

0

005

01

015

02

025

03

035

04

045

Figure 17 Path with the guidance factor

Path compound factor

0

01

02

03

04

05

06

07

08

21

150

04

02

01

03

05

minus03

minus01

minus05

minus04

minus02minus1

minus15

0

005

01

015

02

025

03

035

04

045

Figure 18 Path with the compound factor

map using the VSAPSO optimization path planning methodto solve the multiconstrained optimal path problem in anunstructured environment with stochastic C-GNet modelsThis study draws the following conclusions(1) The traditional environment modeling for the deter-ministic space structure is effective However a complete

description of the cost price stochastic characteristics ofthe nodes and arcs cannot be observed in some networkstructures under the roadway space terrain composed ofroadheader cutting molding and cutting quality influenceHence we proposed a stochastic C-GNet combined with thestochastic network characteristics of the terrain geological


215

105

0minus0508

minus10604minus1502 0 minus2

002040608

1

00501015020250303504045

Figure 19 Cost of the random starting point (02 0)

0

01

02

03

04

05

06

07

08

15 21

minus04 02

01

minus01 05

minus05

minus02 03

04

minus03 0

minus15 minus1

0

005

01

015

02

025

03

035

04

045

Figure 20 Stochastic C-GNet environment model of the random starting point (02 0)

attributes under a narrow and closed nonstructural tunnel-ing(2) The narrow and closed space belonged to theindoor local space category The computational complex-ity was not high and using the particle swarm optimiza-tion algorithm verification environment model was feasi-ble The VSAPSO algorithm used in path planning andoptimized in the convergence speed and accuracy met thetarget planning requirements and the validation of whichin the road in path planning was effective and practi-cal (3) The stochastic constrained modeling method wasbased on the grid network environment and the road-header path planning was completed under a tunnel specialenvironment This modeling and optimizing method wassuitable for the narrow priority large mechanical tunnel-ing working space but not limited to such problems andsolved the problem of intelligent transportation and special

robot autonomous navigation in unstructured environmentdynamic stochastic network optimization problems

Data Availability

The data used to support the findings of this study canbe provided by the corresponding author yangjiannedved163com

Conflicts of Interest

There are no conflicts of interest

Acknowledgments

This work was supported by the National Basic ResearchProgram of China (Grant no 2014CB046306) and the Fun-damental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 800015FC)


All planning paths for multiple targets

0

01

02

03

04

05

06

07

08

15 21

minus04 05

minus05 0

minus03

minus02 02

minus01 01

03

04

minus15 minus1

0

005

01

015

02

025

03

035

Figure 21 All planning path for multi target

005

005

005

005

005

005

005005005

005

005

005

005

005

005

005

005

005 005

005

005

01

01

0101

01

01

01

0101

01

01

01

01

01

0101

0101

01

01

01

01

01

01

01

01

01

01

01

01

01

01

015

015

015

015

015

015

015

015

015

015

015

015

015

015

015015

015

015

015

015

015

015

015

015 015

02

02

02

02

02

02

02

02

02

02

02025 025

025

025

03

03


00

10

2

05

03

04

minus02

minus03

minus04

minus01

minus05 21

15

1

minus1

minus15

0

01

02

03

04

05

06

07

08

0

005

01

015

02

025

03

035

Figure 22 Optimal planning path

References

[1] S-M Feng andG-Q Yin ldquoTransport route optimizationmodelof dangerous goods at the planning levelrdquoHarbinGongyeDaxueXuebaoJournal of Harbin Institute of Technology vol 44 no 8pp 53ndash56 2012

[2] Z Zhou C Li J Zhao and W Xu ldquoA real time path planningalgorithm based on local complicated environmentrdquo HarbinGongye Daxue XuebaoJournal of Harbin Institute of Technologyvol 46 no 8 pp 65ndash71 2014

[3] J Zhuang L Zhang H Sun and Y Su ldquoImproved rapidly-exploring random tree algorithm application in unmanned


surface vehicle local path planningrdquo Harbin Gongye DaxueXuebaoJournal of Harbin Institute of Technology vol 47 no 1pp 112ndash117 2015

[4] F-H Jin H-J Gao and X-J Zhong ldquoResearch on pathplanning of robot using adaptive ant colony systemrdquo HarbinGongye Daxue XuebaoJournal of Harbin Institute of Technologyvol 42 no 7 pp 1014ndash1018 2010

[5] C Chao T Jian and J Zuguang ldquoA path planning algorithm forseeing eye robots based onV-Graph [J]rdquoMechanical Science andTechnology for Aerospace Engineering vol 33 no 4 pp 490ndash495 2014

[6] Y Huan L Naiwen and D Huichuan ldquoResearch on topicalCrawler of T-Graph algorithmrdquo Computer Engineering andDesign vol 35 no 9 pp 3014ndash3017 2014

[7] T Wei Research on path optimization of large scale logisticsvehicle based on Voronoi graph Wuhan University WuhanChina 2013

[8] W Wang and H Wang ldquoReal-time obstacle avoidance tra-jectory planning for missile borne air vehicle based on con-strained artificial potential field methodrdquo Hangkong DongliXuebaoJournal of Aerospace Power vol 29 no 7 pp 1738ndash17432014

[9] Y Jian and Y Zhang ldquoComplete coverage path planningalgorithm for mobile robot Progress and prospectrdquo Journal ofComputer Applications vol 34 no 10 pp 2844ndash2849 2014

[10] R A Conn and M Kam ldquoRobot motion planning on N-dimensional star worlds among moving obstaclesrdquo IEEE Trans-actions on Robotics and Automation vol 14 no 2 pp 320ndash3251998

[11] E Garcia-Fidalgo and A Ortiz ldquoVision-based topologicalmapping and localization methods a surveyrdquo Robotics andAutonomous Systems vol 64 pp 1ndash20 2015

[12] Z Jiandong Robot path planning based on narrow passagerecognition Shanghai Jiao Tong University Shanghai China2012

[13] Y-C Yang J-S Bao and Y Jin ldquoRealistic virtual lunar surfacesimulation methodrdquo Xitong Fangzhen Xuebao Journal ofSystem Simulation vol 19 no 11 pp 2515ndash2518 2007

[14] N Tuygun G Tanir and C Aytekin ldquoRecurrent bacterialmeningitis in children our experience with 14 casesrdquo TheTurkish Journal of Pediatrics vol 52 no 4 pp 348ndash353 2010

[15] S L Laubach and J W Burdick ldquoAutonomous sensor-basedpath-planner for planetary microroversrdquo in Proceedings of the1999 IEEE International Conference on Robotics and Automa-tion ICRA99 pp 347ndash354 May 1999

[16] C HaiyuanThe research on ant colony algorithm of mobile robotpath planning Harbin Engineering University 2013

[17] R Yue and S Wang ldquoPath planning of a climbing robot usingmixed integer linear programmingrdquoBeijingHangkongHangtianDaxue XuebaoJournal of Beijing University of Aeronautics andAstronautics vol 39 no 6 pp 792ndash797 2013

[18] W ZHANG D Zhaolong and J Yang ldquoConstructing methodsfor lunar digital terrainrdquo inAbstracts of the Present Issue vol 25pp 301ndash306 2008

[19] M-P Shi J Wu Y Li and H-G He ldquoA multi-constrainedworldmodelingmethod in lunar rover path-planningrdquoGuofangKeji Daxue XuebaoJournal of National University of DefenseTechnology vol 28 no 5 pp 104ndash118 2006

[20] C Chen C Bu Y He and J Han ldquoEnvironment modeling forlong-range polar rover robotsrdquoChinese Science Bulletin (ChineseVersion) vol 58 no S2 pp 75ndash85 2013

[21] P Urcola M T Lazaro J A Castellanos and L MontanoldquoCooperativeminimum expected length planning for robot for-mations in stochastic mapsrdquo Robotics and Autonomous Systemsvol 87 pp 38ndash50 2017

[22] Y Jianjian T Zhiwei and W Zirui ldquoFault diagnosis on cuttingunit of mine roadheader based on PSO-BP neural networkrdquo inCoal Science and Technology vol 45 pp 129ndash134 10 edition2017

Hindawiwwwhindawicom Volume 2018

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International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


Figure 1 Descartes coordinate grid map

1 1 1124

2322

2120191817

1615

141312

12

11

11

10

10

9

9

8

8

7

7

6

6

6

5

5

5

4

4

4 33

3

3

222

2

Figure 2 Polar coordinate grid map

the floor surface distribution is stochastic The particularityof the walking constraint and the pose adjustment costof the underground roadheader are not reflected in theabove-mentionedmethodsThereforemost of these dynamicstochastic unstructured environment modeling problemscan be more generally classified as a stochastic constraintnetwork

Based on the above this study investigates the stochasticnetwork unstructured spatial environment modeling andavailability problem and increases the high-dimensional con-straint vector and traffic cost in the constraint grid network(C-GNet) nodes and arcs based on grid two-dimensional(2D) terrain data and three-dimensional (3D) altitude dataA network topology with the walking cost weights is con-structed considering the space and the walking constraintsof the roadheader The minimum cost of the network nodesand arcs was determined as the objective function to proposethe variation self-adapting particle swarm optimization algo-rithm and optimize the region path

2 Materials and Methods

21 Environment Model The classical grid network partitionenvironment model mainly includes a Descartes coordinategrid model (Figure 1) and a polar coordinate grid model(Figure 2) which represents the two-dimensional positioninformation119873119890119905 = (119875119870 119865) is a conventional grid network P is aset of nodes representing the grid position point (119909119894 119910119894) K isthe set of altitudes representing the altitude value of the grid

Figure 3 C-GNet diagram structure

position point and F is the set of arcs representing the nodeconnection path

The classical grid network 2D terrain data and the 3Daltitude data cannot reflect other dimensional data such astraffic cost and geological influence cost in the modelingof some stochastic dynamic network environment modelswithout considering the numerical representation of randomdistribution and the behavior constraints of the roadheaderTherefore performing research on the multidimensionalconstraint grid network and traffic cost information is ofpractical significance

Definition 1 The 6-tuple sum = (119875119870 119865119882119883119872) is a gridnetwork system which is the constraint grid network (C-GNet) according to the following structural rules

(1) 119873119890119905 = (119875119870 119865) is a conventional grid network(2) 119870 119875 rarr 119911119894 = 119891(119909119894 119910119894) is the altitude function of

C-GNet(3) W 119865 rarr 119908119894 = 119891(119909119894 119910119894 119911119894 119905119892) is the weight function

of C-GNet(4) X 119865 rarr 119883119894 119894 = 1 2 3 4 is the constraint variant set

of C-GNet(5) MPrarrZ (set of nonnegative integer) is the token or

state

The surface traffic attribute information of the con-strained grid node was established (Figure 3) The proper-ties of the C-GNet structure parameters were defined asProperty 2

Property 2 sum = (119875119870 119865119882119883119872) which satisfies the fol-lowing conditions(1) 119901 isin 119875 is any node ofC-GNet ifexist1199011 1199012 119901119873makingforall1 le 119894 le 119873 Δ119901119894 = 119901 | (119909119894minus1 119910119894minus1)119880(119909119894 119910119894minus1)119880(119909119894+1119910119894minus1)119880(119909119894minus1 119910119894+2)119880(119909119894 119910119894+2) U(119909119894+1 119910119894+2) sub 119875 and Δ119901 = 119901ΔΔ119901 and 119901Δ are the preset and postset of 119901(2) 119891 isin 119865 119891119894119894+1 = 119891(119901119894119894 rarr 119901119894119894+1) represents thedirected arc between (119909119894 119910119894) and (119909119894 119910119894+1)The arc orientationdepends on the starting position of the node120596 represents thearc length weight of C-GNet 120579 represents the arc angle con-straint The state transition (position transfer) between theadjacent nodes is called transition 120595119894minus119895 119872(119901119894) rarr 119872(119901119895)(3) In the C-GNet the path from the source node s tothe target node d cannot be closed

997888997888997888rarr119903119900119886119889 119901119904 120595119904minus119886 119901119886 120595119886minus119887 119901119889997888997888997888rarr1199031199001198861198891(119904 119889)997888997888997888rarr1199031199001198861198892(119904 119889) 997888997888997888rarr119903119900119886119889119899(119904 119889) represent the



119882min (119904 119889) = min( 119889sum119894=119904119895

119908119894120595119894minus119895)

= min( 119889sum119894=119904119895

119891119894119905119903119886V119890119897-119888119900119899119889119894119905119894119900119899120595119894minus119895)(1)

119889sum119894=119904119895

120595119894minus119895 le 10038161003816100381610038161198751199031198991199061198981003816100381610038161003816 minus 1 (2)

2 le 10038161003816100381610038161198751199031198991199061198981003816100381610038161003816 le 119898 + 1 (3)

119875119903119899119906119898 sub 2 3 4 119898 + 1 (4)







sumlowast = (119875119870 119865119882 (119894) 119883119872) (6)



119892 (119894) = |119889 (119894)| times 119886 (7)



119866 = 119873sum119894=1

119892 (119894) (8)


119882119903 = 119873minus1sum119894=1

(119868119899119891119894 + 120579119903119894 + 120579119901119894 + 120579119905119894 + 119871 119894) (9)


119882119903 (119904 119889) = 119882 + 119866 (10)




119909119894119889 (119905 + 1) = 119909119894119889 (119905) + V119894119889 (119905 + 1) (12)

V119894119889 =



V1198941198891003816100381610038161003816V1198941198891003816100381610038161003816 lt Vmax

(13)









119909119894119889 = 119909max times 119903119886119899119889119904 (1 119863) 119903119886119899119889 () ge 119901119909119894119889 119903119886119899119889 () lt 119901 (16)





PSOCFWPSO

APSOVSAPSO

005

115

225

335

4

Fitn

ess v

alue



PSOCFWPSO

APSOVSAPSO

times106

005

115

225

335

4

Fitn

ess v

alue



PSOCFWPSO

APSOVSAPSO

0

20

40

60

80

100

Fitn

ess v

alue






PSOCFWPSO

APSOVSAPSO


0

5

10

15

20

25

Fitn

ess v

alue




Table 2 Variance































215

105

0minus0508

minus10604

minus1502 0 minus2

002040608

1

005

01

015

02

025

03

035

04

045



















0

005

01

015

02

025

03

035

minus1

minus15 0

04

02

01

03

05

minus03

minus01

minus05

minus04

minus02 21

15

0

01

02

03

04

05

06

07

08


002040608











minus15 minus1

PSOCFWPSOVSAPSO

0

01

02

03

04

05

06

07

08

21

150

03

04

02

05

01

minus02

minus01

minus05

minus04

minus03

0

005

01

015

02

025

03

035

04

045


single target path

minus1

minus15 0

04

02

01

03

05

minus03

minus01

minus05

minus04

minus02 21

15

0

01

02

03

04

05

06

07

08

0

005

01

015

02

025

03

035

04

045


4 Conclusions





0

01

02

03

04

05

06

07

08

minus1

minus15 0

04

02

01

03

05

minus03

minus01

minus05

minus04

minus02 21

15

0

005

01

015

02

025

03

035

04

045



0

01

02

03

04

05

06

07

08

21

150

04

02

01

03

05

minus03

minus01

minus05

minus04

minus02minus1

minus15

0

005

01

015

02

025

03

035

04

045





215

105

0minus0508


002040608

1

00501015020250303504045


0

01

02

03

04

05

06

07

08

15 21

minus04 02

01

minus01 05

minus05

minus02 03

04

minus03 0

minus15 minus1

0

005

01

015

02

025

03

035

04

045




Data Availability




Acknowledgments




0

01

02

03

04

05

06

07

08

15 21

minus04 05

minus05 0

minus03

minus02 02

minus01 01

03

04

minus15 minus1

0

005

01

015

02

025

03

035


005

005

005

005

005

005

005005005

005

005

005

005

005

005

005

005

005 005

005

005

01

01

0101

01

01

01

0101

01

01

01

01

01

0101

0101

01

01

01

01

01

01

01

01

01

01

01

01

01

01

015

015

015

015

015

015

015

015

015

015

015

015

015

015

015015

015

015

015

015

015

015

015

015 015

02

02

02

02

02

02

02

02

02

02

02025 025

025

025

03

03


00

10

2

05

03

04

minus02

minus03

minus04

minus01

minus05 21

15

1

minus1

minus15

0

01

02

03

04

05

06

07

08

0

005

01

015

02

025

03

035


References
































Journal of












Journal of







Volume 2018






Volume 2018










119882min (119904 119889) = min( 119889sum119894=119904119895

119908119894120595119894minus119895)

= min( 119889sum119894=119904119895

119891119894119905119903119886V119890119897-119888119900119899119889119894119905119894119900119899120595119894minus119895)(1)

119889sum119894=119904119895

120595119894minus119895 le 10038161003816100381610038161198751199031198991199061198981003816100381610038161003816 minus 1 (2)

2 le 10038161003816100381610038161198751199031198991199061198981003816100381610038161003816 le 119898 + 1 (3)

119875119903119899119906119898 sub 2 3 4 119898 + 1 (4)







sumlowast = (119875119870 119865119882 (119894) 119883119872) (6)



119892 (119894) = |119889 (119894)| times 119886 (7)



119866 = 119873sum119894=1

119892 (119894) (8)


119882119903 = 119873minus1sum119894=1

(119868119899119891119894 + 120579119903119894 + 120579119901119894 + 120579119905119894 + 119871 119894) (9)


119882119903 (119904 119889) = 119882 + 119866 (10)




119909119894119889 (119905 + 1) = 119909119894119889 (119905) + V119894119889 (119905 + 1) (12)

V119894119889 =



V1198941198891003816100381610038161003816V1198941198891003816100381610038161003816 lt Vmax

(13)









119909119894119889 = 119909max times 119903119886119899119889119904 (1 119863) 119903119886119899119889 () ge 119901119909119894119889 119903119886119899119889 () lt 119901 (16)





PSOCFWPSO

APSOVSAPSO

005

115

225

335

4

Fitn

ess v

alue



PSOCFWPSO

APSOVSAPSO

times106

005

115

225

335

4

Fitn

ess v

alue



PSOCFWPSO

APSOVSAPSO

0

20

40

60

80

100

Fitn

ess v

alue






PSOCFWPSO

APSOVSAPSO


0

5

10

15

20

25

Fitn

ess v

alue




Table 2 Variance































215

105

0minus0508

minus10604

minus1502 0 minus2

002040608

1

005

01

015

02

025

03

035

04

045



















0

005

01

015

02

025

03

035

minus1

minus15 0

04

02

01

03

05

minus03

minus01

minus05

minus04

minus02 21

15

0

01

02

03

04

05

06

07

08


002040608











minus15 minus1

PSOCFWPSOVSAPSO

0

01

02

03

04

05

06

07

08

21

150

03

04

02

05

01

minus02

minus01

minus05

minus04

minus03

0

005

01

015

02

025

03

035

04

045


single target path

minus1

minus15 0

04

02

01

03

05

minus03

minus01

minus05

minus04

minus02 21

15

0

01

02

03

04

05

06

07

08

0

005

01

015

02

025

03

035

04

045


4 Conclusions





0

01

02

03

04

05

06

07

08

minus1

minus15 0

04

02

01

03

05

minus03

minus01

minus05

minus04

minus02 21

15

0

005

01

015

02

025

03

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04

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0

01

02

03

04

05

06

07

08

21

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04

02

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215

105

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002040608

1

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15 21

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04

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0

005

01

015

02

025

03

035

04

045




Data Availability




Acknowledgments




0

01

02

03

04

05

06

07

08

15 21

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minus05 0

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03

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03

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06

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015

02

025

03

035


References
































Journal of












Journal of







Volume 2018






Volume 2018















119909119894119889 = 119909max times 119903119886119899119889119904 (1 119863) 119903119886119899119889 () ge 119901119909119894119889 119903119886119899119889 () lt 119901 (16)





PSOCFWPSO

APSOVSAPSO

005

115

225

335

4

Fitn

ess v

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PSOCFWPSO

APSOVSAPSO

times106

005

115

225

335

4

Fitn

ess v

alue



PSOCFWPSO

APSOVSAPSO

0

20

40

60

80

100

Fitn

ess v

alue






PSOCFWPSO

APSOVSAPSO


0

5

10

15

20

25

Fitn

ess v

alue




Table 2 Variance































215

105

0minus0508

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1

005

01

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15

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PSOCFWPSOVSAPSO

0

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04

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15

0

01

02

03

04

05

06

07

08

0

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02

025

03

035

04

045


4 Conclusions





0

01

02

03

04

05

06

07

08

minus1

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04

02

01

03

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215

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04

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0

005

01

015

02

025

03

035

04

045




Data Availability




Acknowledgments




0

01

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05

06

07

08

15 21

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minus05 0

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02

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07

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01

015

02

025

03

035


References
































Journal of












Journal of







Volume 2018






Volume 2018









PSOCFWPSO

APSOVSAPSO


0

5

10

15

20

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Fitn

ess v

alue




Table 2 Variance































215

105

0minus0508

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002040608

1

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002040608











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15

0

01

02

03

04

05

06

07

08

0

005

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02

025

03

035

04

045


4 Conclusions





0

01

02

03

04

05

06

07

08

minus1

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04

02

01

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15

0

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0

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215

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002040608

1

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04

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0

005

01

015

02

025

03

035

04

045




Data Availability




Acknowledgments




0

01

02

03

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05

06

07

08

15 21

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minus05 0

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03

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02

02

02

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02

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03

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06

07

08

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02

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035


References
































Journal of












Journal of







Volume 2018






Volume 2018

























215

105

0minus0508

minus10604

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002040608

1

005

01

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02

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15

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002040608











minus15 minus1

PSOCFWPSOVSAPSO

0

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08

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0

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single target path

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04

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15

0

01

02

03

04

05

06

07

08

0

005

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02

025

03

035

04

045


4 Conclusions





0

01

02

03

04

05

06

07

08

minus1

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04

02

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0

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0

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215

105

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0

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01

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045




Data Availability




Acknowledgments




0

01

02

03

04

05

06

07

08

15 21

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minus05 0

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03

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02

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0

01

02

03

04

05

06

07

08

0

005

01

015

02

025

03

035


References
































Journal of












Journal of







Volume 2018






Volume 2018












0

005

01

015

02

025

03

035

minus1

minus15 0

04

02

01

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15

0

01

02

03

04

05

06

07

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002040608











minus15 minus1

PSOCFWPSOVSAPSO

0

01

02

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single target path

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04

02

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minus02 21

15

0

01

02

03

04

05

06

07

08

0

005

01

015

02

025

03

035

04

045


4 Conclusions





0

01

02

03

04

05

06

07

08

minus1

minus15 0

04

02

01

03

05

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minus01

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15

0

005

01

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02

025

03

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045



0

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06

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08

21

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04

02

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0

005

01

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02

025

03

035

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045





215

105

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002040608

1

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0

01

02

03

04

05

06

07

08

15 21

minus04 02

01

minus01 05

minus05

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04

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0

005

01

015

02

025

03

035

04

045




Data Availability




Acknowledgments




0

01

02

03

04

05

06

07

08

15 21

minus04 05

minus05 0

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03

04

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0

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01

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02

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03

035


005

005

005

005

005

005

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005

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005

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005 005

005

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0101

01

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02

02

02

02

02

02

02

02

02

02

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03

03


00

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2

05

03

04

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15

1

minus1

minus15

0

01

02

03

04

05

06

07

08

0

005

01

015

02

025

03

035


References
































Journal of












Journal of







Volume 2018






Volume 2018









minus15 minus1

PSOCFWPSOVSAPSO

0

01

02

03

04

05

06

07

08

21

150

03

04

02

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01

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minus01

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0

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01

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02

025

03

035

04

045


single target path

minus1

minus15 0

04

02

01

03

05

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minus01

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minus02 21

15

0

01

02

03

04

05

06

07

08

0

005

01

015

02

025

03

035

04

045


4 Conclusions





0

01

02

03

04

05

06

07

08

minus1

minus15 0

04

02

01

03

05

minus03

minus01

minus05

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minus02 21

15

0

005

01

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02

025

03

035

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0

01

02

03

04

05

06

07

08

21

150

04

02

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05

minus03

minus01

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minus15

0

005

01

015

02

025

03

035

04

045





215

105

0minus0508


002040608

1

00501015020250303504045


0

01

02

03

04

05

06

07

08

15 21

minus04 02

01

minus01 05

minus05

minus02 03

04

minus03 0

minus15 minus1

0

005

01

015

02

025

03

035

04

045




Data Availability




Acknowledgments




0

01

02

03

04

05

06

07

08

15 21

minus04 05

minus05 0

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minus02 02

minus01 01

03

04

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0

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005

005

005

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005

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01

01

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02

02

02

02

02

02

02

02

02

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00

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2

05

03

04

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15

1

minus1

minus15

0

01

02

03

04

05

06

07

08

0

005

01

015

02

025

03

035


References
































Journal of












Journal of







Volume 2018






Volume 2018










0

01

02

03

04

05

06

07

08

minus1

minus15 0

04

02

01

03

05

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minus01

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15

0

005

01

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02

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04

045



0

01

02

03

04

05

06

07

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21

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04

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0

005

01

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02

025

03

035

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045





215

105

0minus0508


002040608

1

00501015020250303504045


0

01

02

03

04

05

06

07

08

15 21

minus04 02

01

minus01 05

minus05

minus02 03

04

minus03 0

minus15 minus1

0

005

01

015

02

025

03

035

04

045




Data Availability




Acknowledgments




0

01

02

03

04

05

06

07

08

15 21

minus04 05

minus05 0

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minus02 02

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03

04

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0

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005

005

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15

1

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minus15

0

01

02

03

04

05

06

07

08

0

005

01

015

02

025

03

035


References
































Journal of












Journal of







Volume 2018






Volume 2018









215

105

0minus0508


002040608

1

00501015020250303504045


0

01

02

03

04

05

06

07

08

15 21

minus04 02

01

minus01 05

minus05

minus02 03

04

minus03 0

minus15 minus1

0

005

01

015

02

025

03

035

04

045




Data Availability




Acknowledgments




0

01

02

03

04

05

06

07

08

15 21

minus04 05

minus05 0

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03

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0

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stochastic c-gnet environment modeling and path planning...

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