review article survey of robot 3d path planning...

Review ArticleSurvey of Robot 3D Path Planning Algorithms

Liang Yang12 Juntong Qi1 Dalei Song1 Jizhong Xiao3 Jianda Han1 and Yong Xia4

1Shenyang Institute of Automation Nanta 114th Street Shenhe District Shenyang 10016 China2University of Chinese Academy of Sciences 19 Yuquan Road Beijing 100049 China3The City College City University of New York Convent Avenue at 140th Street New York NY USA4State Grid Liaoning Electric Power Company Benxi Liaoning 117000 China

Correspondence should be addressed to Jizhong Xiao jxiaoccnycunyedu

Received 25 November 2015 Accepted 3 April 2016

Academic Editor Petko Petkov

Copyright copy 2016 Liang 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

Robot 3D (three-dimension) path planning targets for finding an optimal and collision-free path in a 3D workspace while takinginto account kinematic constraints (including geometric physical and temporal constraints)The purpose of path planning unlikemotion planning which must be taken into consideration of dynamics is to find a kinematically optimal path with the least time aswell asmodel the environment completelyWe discuss the fundamentals of thesemost successful robot 3Dpath planning algorithmswhich have been developed in recent years and concentrate on universally applicable algorithmswhich can be implemented in aerialrobots ground robots and underwater robots This paper classifies all the methods into five categories based on their exploringmechanisms and proposes a category called multifusion based algorithms For all these algorithms they are analyzed from atime efficiency and implementable area perspective Furthermore a comprehensive applicable analysis for each kind of methodis presented after considering their merits and weaknesses

1 Introduction

Advances in commercial grade technology and advancedresearchmake it possible for robots to appear in everyday lifeModern robots of different varieties include industrial robotsservice robots have seen bright future For robots such asGoogle Self-Driving Car [1] and iRobot Vacuum Cleaningrobot [2] one of themost basic and important abilities is pathplanning that is autonomous routing This requires bothtime efficiency to react to any emergency situation and safetyconsideration for implementation

Path planning targets for moving robots from their initiallocations to the goal location by their own actuators andstrategies and during the process robots must always beable to avoid obstacles to maintain safety Robots such asunderwater robots [3 4] wall-climbing robot [5] and microair vehicles [6ndash8] have already been tested purposely withdifferent kinds of methods These methods can be basi-cally perceived from [9ndash12] where algorithms were alreadydetailed synthesized These works contributes a lot howeverthey are analyzed in a general view without comparison and

analyzed in specific perspectives as well as covering the latestworksThus we need to provide a comprehensive analysis on3D path planning methods

By reviewing the latest works it is not hard to see thatmost works concentrate only on two dimensions (2D) thuslimiting the behaviors of the robots only to surface or eachiteration considering the height as a constant to achieve a25-dimensional (25D) method Choset [9] summarized alot of his own and othersrsquo works however he mainly con-centrates on 2D environment and therefore algorithms likebioinspired algorithms were paid no attention LaValle et al[10] focused mainly on sampling based algorithms Noneof these researchers had analyzed all the algorithms in 3Dplanning area Surveys such as [13] almost all analyze the pathplanning problems under 2D condition Reference [14] onlyfocuses on micro air vehiclersquos path planning problems andalgorithms likemathematic optimal algorithmswere ignoredLately authors in [13] did a thorough survey on coverage pathplanning which is classified by distinguishing decomposi-tion methods They provided a basic understanding of 3Dpath planning methods and then narrowed their discussion

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016 Article ID 7426913 22 pageshttpdxdoiorg10115520167426913

2 Journal of Control Science and Engineering

(a) Forest [15] (b) Urban [16] (c) Underwater [17]

Figure 1 Examples of 3D complex environments

to sampling based algorithm to explain the 3D planningidea

Facing the more and more challenging environment thatrobots faced where environments tend to be unstructuredand full of uncertain factors 3D path planning algorithmsare urgently needed nowadays Several typical kinds of 3Denvironments are presented in Figure 1 including foresturban and underwater environments When planning inthese complex situations a simple 2D algorithm will not bequalified thus 3D path planning algorithms are needed

Path planning in 3D environment shows great prospectbut unlike 2D path planning the difficulties increase expo-nentially with kinematic constraints In order to plan acollision-free path through the cluster environment theproblem of how to model the environment while takingthe kinematic constraints into consideration needs to besolved From the optimization point of view finding a 3Doptimal path planning problem is NP-hard thus there existno common solutions

3D path planning algorithms include visibility graph [18]which works by connecting visible vertexes of polyhedronrandom-exploring algorithms such as rapidly exploring ran-dom tree [19] Probabilistic Road Map [20] optimal searchalgorithms (such as Dijkstrarsquos algorithm [21] Alowast [22] andDlowast [23]) and bioinspired planning algorithmsWhatmust beemphasized is that this paper only pays attention to broadlyapplicable methods proposed Algorithms such as manifoldbased algorithms [24] which generate smooth path havebeen ignored due to reason that it is only applicable torigid body robots without aerodynamical or hydromechan-ical influences Synthesizing all these methods a two-stepprocedure of 3D path planning is summarized as follows

Step 1 The first step is environment perception and model-ing usually using a grid map [25] (with occupancy proba-bility) or a convex or nonconvex region in combination ofpolynomial form [26]

Step 2 Then path planning algorithm is employed to find thebest path according to the cost function with the ability toachieve both time efficiency and cost minimum

Steps above are not absolute procedures we must obeyand Step 1 can be divided into two parts or Step 2 integrateswith Step 1 According to Step 2 this paper discusses theefficiency of each algorithm and particularly concentrateson time complexity and applicability Path planning in 3Denvironments may face much more uncertainties thus allthis should be taken into consideration to achieve the best

path Local minimal and global optimal are two contradictivepoints this paper will analyze these two properties of thealgorithms This paper mainly answered several importantquestions (a) what is the taxonomy of these 3D path planningalgorithms (b) How do these algorithms work in order tofind the path (c) Why is it suitable or not suitable for suchenvironment

This paper is an extension of [27] with more comprehen-sive explanation on each kind of algorithms

The main contributions of this paper are listed below(1)Thefirst contribution is proposing a taxonomymethod

for 3D path planning algorithms and puts forward a newcategory called multifusion based algorithms

(2) The second contribution of this paper is providing acomprehensive analysis of 3D path planning algorithmwhichcontains almost all of the current most successful methods

The following sections are arranged as follows Sec-tion 2 discusses some controversial points which need tobe declared and gives definitions to these problems forfurther discussion Section 3 explains the taxonomy of 3Dpath planning algorithms and gives a detailed analysis ofthe taxonomyrsquos reason and also lists elements of each cate-gory This section includes the basic concept of each kindof algorithms A series of sampling based algorithms arediscussed particularly and each element is put forward bycomparison in Section 4 Node based optimal algorithmsrsquocommon properties are analyzed in Section 5 and thiskind of algorithms shares the same merits In Section 6this paper discusses a special kind of planning algorithmsthat is mathematic model based algorithms and confirmsit to be path planning algorithm Section 7 concentrateson the working mechanism of bioinspired algorithms andthree typical algorithms are analyzed elaborately Multifusionbased algorithms are discussed in Section 8 which arecommonly ignored Some typical researches are listed toprove this categoryThe last section outlines a conclusion andsome directions for further research

2 Preliminary Materials

This section discusses some controversial or ignored pointswhich needed to be declared for further discussion inthe following sections namely the difference between pathplanning and optimal path planning and definition of pathplanning and trajectory planning

21 Problem Statement Robots have the ability to operatewithout (or just a little guiding from) human Ranging from

Journal of Control Science and Engineering 3

3D path planning algorithms

Samplingbased

algorithms

basedNode

optimalalgorithms

modelMathmtic

basedalgorithms

Bioinspiredalgorithms

Multifusionbased

algorithms

Figure 2 3D path planning taxonomy

underwater robots to aerial robots when facing outdooror indoor complex situations they need a path planner todetermine their next step movement For path planning thedefinition varies according to [8ndash10 57] this paper presentsa more canonical definition based on [57]

Robots are assumed to operate in a three-dimension (1198773)space sometimes called the workspace 119908 This workspacewill often contain obstacles let 119908119900

119894be the 119894th obstacle The

free workspace without threat of obstacles is the set of points119908free = 119908cup119894 119908119900119894 it is the spacewhere the robots should alwaysstay For robots the initial point 119909init is an element of 119908freeand the goal region 119909goal is also an element of119908freeThus pathplanning problem is defined by a triplet (119909init 119909goal 119908free) andthe following definitions are given

Definition 1 (path planning) Given a function 120575 [0 119879] rarr

1198773 of bounded variation where 120575(0) = 119909init and 120575(119879) = 119909goal

if there exists a process Φ that can achieve 120575(120591) isin 119908free forall 120591 isin [0 119879] the process Φ must be a continuous processwithout break thenΦ is called path planning

Definition 2 (optimal path planning) Given a path plan-ning problem (119909init 119909goal 119908free) and a cost function 119862

sum rarr 119877 ge 0 (sum denote the set of all paths) ifDefinition 1 is fulfilled to find a path 120575

1015840 and 119862(1205751015840

) =

min(119862(120575) 120575 is the set of all feasible paths) then 1205751015840 is the

optimal path and Φ1015840 is optimal path planning

22 Path Planning and Trajectory Planning Path planningand trajectory planning problems are two distinct parts ofrobotics but they are intimately related There exist severalworks [8 9 11] concerned about this problem Synthesizing allthe corresponding knowledge together this paper supportsthe following definitions for further discussion

Definition 3 (path planning (augmentation)) Find a contin-uous curve (no need to be smooth) in the configuration spacethat begins from the start node 119909init to the goal end node119909goal and the curve should obey the criteria (a) beingwithouttime continuous consideration (b) including stops in definedposition and (c) beingmade up of a number of segments andeach can be a trajectory

Definition 4 (trajectory planning) Trajectory planning usu-ally refers to the problem of taking the solution from a robotpath planning algorithm and determining how tomove alongthe path Trajectory is a set of states that are associated withtime it can be described mathematically as a polynomial119883(119905) and velocities and accelerations can be computed bytaking derivatives with respect to time It considers thekinodynamic constraints which can be an element of path

3 3D Path Planning Algorithm Taxonomy

Algorithms of 3D path planning have been arising since lastcentury methods have different characteristics and can beapplied to different robots and environments For researchersand engineers being stunned to swim in the algorithm sea is acommon scene to start in this fieldThis paper reviews almostall the representative works and books and sorts out almostall the successful 3D path planning methods such as rapidlyexploring random trees (RRT) Probabilistic Road Maps(PRM) Artificial Potential Field [86] and Mixed-IntegerProgramming [87] The taxonomy that proposed classifyingcurrent approaches of 3D path planning is illustrated inFigure 2

This paper divides 3D path planning algorithms into fivecategories the categories are distinguished from each otherby their unique properties such as sampling based algo-rithms explored by applying Monte Carlo sampling (or likelyapproaches) to achieve visible connection From Sections 4ndash8an exhaustive discussion of each category is presented

31 Sampling Based Algorithms Elements and Analysis Thiskind of methods needs some preknown information ofthe whole workspace that is a mathematic representationto describe the workspace This kind usually samples theenvironment as a set of nodes or cells or in other formsThenmap the environment or just search randomly to achieve afeasible path The elements of sampling based algorithms areillustrated in Figure 3

For ETC it means the improved or similar versions ofcorresponding algorithms DDRRT isDynamicDomain RRTalgorithm There is no doubt that rapidly exploring random


Sampling based algorithms

Active Passive

RRTlowast

RRTDDRRT

and so forth

3DVoronoi

RRGand so forth

RMlowast) and

PRMk-PRMs-PRM(P

so forth

Artificial

Fieldand so forth

Potential

Figure 3 Elements of sampling based algorithms

Node based optimal algorithms

Dijkstrarsquosalgorithm

and so forth

Dlowast

and so forthDlowast-LiteThetalowast

Alowast LPAlowast

lowastlazy Thetaand

Figure 4 Elements of node based algorithms

trees (RRT) andProbabilistic RoadMaps (PRM) are samplingbased algorithms For 3D Voronoi it forms a 3D obstaclefree network based on preknown knowledge of the wholeenvironment which contains obstacle and free regions Toachieve this network Voronoi algorithm still works in anoptimization way to follow the varying edges to ensure equaldistance [88] Artificial Potential Field algorithms are sortedas sampling based algorithm due to the fact that it needs thewhole workspace sampling information to escape from localminima

It is shown in the second level of Figure 3 that this paperdivides the sampling based algorithms into twomore detailedparts active and passive Active means algorithm such asrapidly exploring random trees which can achieve the bestfeasible path to the goal all by its own processing procedurePassive means algorithms such as Probabilistic Road Maps(PRM) only generate a road net from start to the goal thus acombination of search algorithms to pick up the best feasiblepath in the net map where many feasible paths exist Thispaper classifies the algorithms which cannot independentlyfind the best path or any single navigation path as passive

32 Node Based Optimal Algorithms Elements and Anal-ysis Node based optimal algorithms explore through the

decomposed graph [25] Analyzing from the point of thesearch mechanism node based optimal algorithms sharethe same property that they explore among a set of nodes(cell) in the map where information sensing and processingprocedures are already executed This kind of methodscan always find an optimal path according to the certaindecomposition Figure 4 illustrates the typical elements ofnode based optimal algorithms

Where LPAlowast [48] represents Lifelong Planning Alowast algo-rithm which is a repeating version of Alowast (ie having theability to handle dynamic threats) Dlowast-Lite [56] was proposedbased on LPAlowast which deals with dynamic threat situationsDijkstrarsquos algorithmsAlowast andDlowast are traditionally classified asdiscrete optimal planning [10] or road map algorithms [12]or search algorithms [9] and so on However they do notcollide with each other and just express the same thing wherealgorithms like this kind deal with discrete optimizationbased on grid decomposition

33 Mathematic Model Based Algorithms Elements andAnalysis Mathematic model based algorithms include linearalgorithms and optimal control These methods model theenvironment (kinematic constraints) as well as the system(dynamic) and then bound the cost functionwith all the kine-matic and dynamic constraints bounds which are inequalitiesor equations to achieve an optimal solution The elementsof mathematic model based algorithms are presented inFigure 5

Where flatness based method was first proposed byChamseddine et al [63] this method employs differentialflatness to ensure control flatness along the reference pathit linearizes the nonlinear kinodynamic constraints to forma rather simple form MILP is Mixed-Integer Linear Pro-gramming [5] which has a strong modeling capability todescribe almost all the information BIP is Binary LinearProgramming [62] it is a special case of linear programmingwhere the variables have only 0-1 integer value


Linear algorithms

NLP

Optimal control

NLP

and so forthFlatness based

Mathematic modelbased algorithms

and soMILP

forth

BIPand soforth

Figure 5 Elements of mathematic model based algorithms

Neural network

GAand soforth

SFLAand soforth

Evolutionary algorithms

and soforth

MAand soforth

PSOand soforth

ACO


Figure 6 Elements of bioinspired algorithms

34 Bioinspired Algorithms Elements and Analysis Bioin-spired algorithms originate from mimicking biologicalbehavior to solve problems This kind of planning methodsleaves out the process of constructing complex environmentmodels to search a near optimal path based on stochasticapproaches it overcomes the weakness where general math-ematic model based algorithms often fail (or drop into localminimum) in solving NP-hard problem with large numberof variables and nonlinear objective functions [4] Figure 6presents a set of typical current methods belonging to thiscategory

Where GA is Genetic Algorithm [89] it is the mostfamous population-based numerical optimization methodMA represents Memetic Algorithm [90] it is a population-based heuristic exploring approach for combinatorial opti-mization problems PSO is particle swarm optimization[91] it is a population-based stochastic optimization algo-rithm ACO is Ant Colony Optimization [92] that imitatesthe behavior of ant in finding the shortest path by usingpheromone information and SFLA is Shuffled Frog LeapingAlgorithm [93]which is the combination ofMAandPSO Forbioinspired algorithms they are divided into EvolutionaryAlgorithm (EA) and Neural Network (NN) due to thefact that they are analyzed at different level which will bediscussed in detail in the following sections

35 Multifusion Based Algorithms Analysis Fusion is a latelyarising approach to improving the performance of 3D path

planning algorithms algorithms can benefit each other bythis way Usually algorithms tend to fuse in a layer bylayer way and aim to plan an optimal path (with betterreal time or nonlocal optimal performance) For exampleArtificial Potential Field algorithms usually tend to drop intolocal minima without navigation function or other tricksProbabilistic Road Maps also cannot generate an optimalsingle path by itself Thus this paper classifies this kindof algorithms which are introduced by combining severalalgorithms together to achieve a better performance asmultifusion based algorithms Section 8 will give a canonicalillustration to this category

4 Sampling Based Algorithms

In Section 31 a list of sampling based 3D path planningalgorithms is already illustrated and this part aims to givea detailed analysis of this kind of algorithms In order togive a clear image of this kind of algorithms this paperdefines the RRT and its improved versions as RRT series andPRM series contains PRM and its improved versions Thisdefinition is also applicable toVoronoi andArtificial PotentialField

41 RRT Series

411 Rapidly Exploring Random Trees Rapidly exploringrandom tree (RRT) method is first proposed by LaValle[19] The method attempted to solve path planning prob-lems under holonomic nonholonomic and kinodynamicconstraints RRT has the advantage of handling multi-DOFproblems thus it is widely used for PR2 and other robots [94]Authors in [31] proposed a fast local escaping version calledDynamic Domain RRT which will be analyzed below Kara-man and Frazzoli [30] solved nonoptimal results problem ofRRT by introducing RRG and a heuristic method RRTlowast

RRT rapidly searches the configuration space to generatea path connecting the start node and the goal node In eachstep a new node is sampled if the extension from the sampledto the nearest node succeeds a new node will be addedWhen this kind of method is applied to 3D environment itnormally assumes that there exists a 3D configuration space119883 = 119862 The configuration space consists of two parts afixed obstacle region 119883obs sub 119883 which must be avoided andan obstacle free region 119883free sub 119883 where the robots muststay Corresponding to the configuration space a path state(or vertex) set 119875 includes all the sampling vertices which aregenerated by RRT exploration process In order to implementthe algorithm the following steps should be obeyed (see inFigure 7)

Step 1 First add the initial state 119909init isin 119883free in 119875 as the firstvertex Then randomly choose a state 119909random in 119883free andFigure 7 illustrates two states which are 119909random1 and 119909random2

Step 2 Select a nearest state 119909near to the newly generated state119909random in 119875 based on a certain metric (mostly Euclideanmetric) which is already designed regard 119909near as the parentstate of 119909random


xinit

xrandom1

xrandom2

xnew1

xnew2xnear

120574

Figure 7 Exploring procedure of RRT algorithms The cyan circlesrepresent obstacle regionswhich cannot be passed 120574 ismaximal stepextending length according to constraints and cost function stated

Step 3 119909random is the state which shows the direction wherethe next step should go but may be beyond the robotrsquosreachabilityThus a control input factor is added consideringthe kinodynamic constraints in a cost function120601 = 119891(119909 119910 119911)form Then according to the constraints 119903 and cost function120601 we get the reachable state 119909new judging 119909new whether it isin119883free If it locates in119883free then add it to the path set 119875 elseignore this state It is illustrated in Figure 7 we delete 119909new1and repeat the whole process

Although RRT can find a path to the goal it is still theproblem that RRT explores based on Monte Carlo randomsampling which is always biasing explored region as it willincrease with the time The method will consume much timeto find a way out when the environments are cluttered letalone converge to the optimal

412 Dynamic Domain RRT Dynamic Domain RRT(DDRRT) aims to solve the shortcoming where RRTconsiders none of the obstacles region in the configurationspace especially local trap region RRT does not considerthe information of local environments and thus causesinappropriate sampling which may lead to low timeefficiency An intuitive comparison is given in Figure 8where the red arrow means the guiding to possible nextextending direction

DDRRT and RRT differ in Step 1 DDRRT introducesa 119899-dimensional sphere (based on the dimension of theenvironment) of certain radius 119903 at the center 119909near torepresent the reachable region At each time if the distanceof new sampling node 119909random to the nearest node 119909near iswithin radius 119903 the sampling node will be chosen otherwiseit will be neglected Then try to connect 119909near and 119909random if

the extension is succeed then set 119903 = infin otherwise set 119903 = 119877where 119877 is the boundary radius of dynamic domain

DDRRT solves the Voronoi bias problem of general RRTit can ensure fast exploration However it is the same asRRT where no post-smooth-processing is included the pathwhich is generated by DDRRT will never tend to be optimal

413 Rapidly Exploring Random Graph (RRG) RRT per-forms well in practice and can guarantee completeness butit pays almost no attention to the quality of the resultsand it is proved that RRT algorithms are not asymptoticallyoptimal [30] To ensure asymptotic optimality authors in [30]introduced rapidly exploring randomgraph (RRG)Theworkused a structure 119896 = (119878 119878

0 119879 119871) to represent the performance

of the system where 119878 is the set of states 1198780sube 119878 is the initial

states sets 119879 sube 119878 times 119878 is the transition relation and 119871 is thelabeling function which is used to map each state to the set ofatomic propositions

It is illustrated in Figure 9 where ldquocurrent new noderdquodenotes the new nodes generated at current steps UnlikeRRT RRG tries to extend to every state returned by Near()(line 7 of Algorithm 2) which is illustrated in Figure 9 InFigure 9(a) |119901| is the radius stated to choose the neighborstates around the center of ldquocurrent new noderdquo then 119909newconnects all these nodes to form a graph as illustrated inFigure 9(b) if the connection is collision-free (in Algorithm 2from line 7 to line 12)The pseudocode is given in Algorithms1 and 2

Line 7 shows the main difference between RRT and RRGwhere RRG also connects to satisfaction vertices which liein the circle region thus it forms a graph RRG holds theadvantage of extending all the vertices returned by the Near()process and then connects them to present a more complexmap Although it seems to be more complex in some way italmost surely can converge to an optimal path Reference [29]showed experimental results of RRG and the comparisonto PRM proves the advantage of RRG Also the conclusionsuggested that RRG enables navigatingmultiple robots simul-taneously However it generates a complex network like PRMand thus cannot find the optimal path by itself

414 RRT-Star (119877119877119879lowast) RRTlowast [30] is the tree version ofRRG which also preserves the asymptotic optimal propertyas RRG it is proposed to tackle differential constraints RRTlowastremoves probable bad connection which works in a refiningway thus it optimizes the solutions to be less expensive thanthey used to be Here a cost-function cost(119901) is defined torepresent the cost of the unique path from 119909init to an arbitrarystate 119901 isin 119875 and gives the cost(119909init) equal to zero initiallyRRTlowast differs RRG in postprocessing process compared toAlgorithm 2 RRTlowast pseudocode is given in Algorithm 3 withintuitive illustration Figure 10

Let us take the same example case as illustrated inFigure 9(a) RRTlowast first finds the nearest state as well asthe neighbor states 119883near and then adds the nearest state tothe tree as well as the minimal cost connection if it exists(from line 4 to line 12) Further RRTlowast tries to eliminate theconnections which have a larger cost by connection via 119909newstate (from line 14 to line 21) Because of the pruning and


xinit

(a) It is hard for general RRT to deal with bug trap problem

xinit

(b) DDRRT tackles trap problem by limiting the extension nearthe margin

Figure 8 DDRRT compares with general RRT

Current new node |P|

xinit

(a) The near nodes returned

xinit

(b) 119909new connects all near nodes to form RRG

Figure 9 Typical operation of RRG

reconnecting the tree turns to bemore compact and dense asillustrated in Figure 10(b) Consequently the overall minimalcost can be obtained However the whole time consumptionincreases and also it cannot work to generate multipathChoudhury et al [28] assumed that if the nearest parentalready has children nearby then the second best parentwill be chosen This method solves the problem of generalRRT and RRTlowast by supporting an on-line fast replanningmethod

42 PRM Series Unlike RRT Probabilistic RoadMap (PRM)considers different choices for the set of states to whichconnections are attempted PRM [20] is the first popularmultiple-query method for building a road map by usingsampling approachWhen applied to 3D space it first definesthe configuration space 119883 and an obstacle free space 119883freeThe method samples a random state in 119883free Then it triesto connect to nearest 119896 neighbors (119870-PRM) or connect tostates within a 120575-ball region or connect to states under


x1

x2

x3

xnearest

xinit

(a) Pruning process of RRTlowast

x1

x2

x3

xinit

(b) The final RRTlowast graph

Figure 10 RRTlowast postrefining process that is pruning of heavy connect

119864 is the set of edges(1) 119881 larr 119909init(2) for 119894 = 0 to 119899(3) 119866 = (119875 119864)

(4) 119909rand larr 119894th randomly chosen state(5) (119901 119864) = ExploreRRG(119866 119909rand)(6) End

Algorithm 1 Body of RRG

computational burden which is a combination of the abovetwo that is 119870-PRM cap 120575-ball It should be declared that eachconnection is collision-free After the road map is absolutelyformed a node based search algorithm (Dijkstra Alowast Dlowastetc) is used to find the least cost path from initial stateto the goal state The pseudocode of PRM is illustrated inAlgorithm 4

In Algorithm 4 PRM adds all the connections which arewithout collision (from line 4 to line 9) Authors in [95] firstimplemented PRM into 3D environment and showed fastexploration performanceHowever with the expanding of theexploring graph the expense on collision checking increasesAmato et al [32] put it forward by proposing an obstacle-based nodes generation strategy at each step the road mapcandidate points are selected on the obstacle surface Thistrick reduces the total processing time by ignoring theuseless points in obstacle region Hsu et al [33] solved theldquodynamic threat weakrdquo problem of PRM by introducing aconcept of ldquomilestonerdquo that is state times time which createsa real time graph connecting the initial point and the goalpoint It was implemented in ground robots and proved ofhaving quick convergence ability Yan et al [6] proposed anoctree to build 3D grid-structure which weakens the effectof randomness thus guaranteeing fast searching ability The

Table 1 Elements of PRM

Name of eachsub-PRM Criteria of corresponding algorithm

119870-PRM Meaning each step choose nearest 119896 neighbors tobe the states which are under consideration

119878-PRMThe expected states are to be chosen within a ball(or circle) region for all states included in theradius 119903 will be chosen to connect to the vertex

119870-119878-PRMA combination of above two where for a givenradius 119903 the upper bound of the vertices to bechosen is 119896

work gave a great thought of random sampling in boundingbox array Reference [30] introduced PRMlowast to guaranteeasymptotical optimality Authors in [34] concentrated on theproblem that typical PRM cannot tackle dynamic threatsthey imported the idea of potential to avoid this which iscalled Reactive Deformation Road Maps (RDR)

PRM has the bottleneck of not being able to determinean efficient near vertex choosing and connection principleBased on the difference of vertex chosen criteria this papercombines the idea of [10 57] and divides PRM into three partsin Table 1

Although 119870-PRM has the advantage of adopting enoughsamples to ensure smoothness by adjusting parameter 119896 theextension will be biased if a fixed direction tends to be muchmore dense 119878-PRM can keep out the shortcoming of 119870-PRM but an unreasonable radius 119903may result in an excessivecomputational burden 119870-119878-PRM likes an adaptable 119878-PRMit can guarantee all direction connection and also limit thecomputational burden to a certain degree

43 Voronoi Shamos and Hoey [96] introduced the Voronoidiagram into the field of computational geometry it is firstly


Steer(119909 119910) returns a point closer to 119910 but within the reach of 119909 by a certain valueNear(119866 119909rand |119875|) return all vertices in a ball center at 119909new and radius is 119903

119899

Freeobsta(119909 119910)means no obstacle between 119909 119910(1) 119875

1015840

= 119875 1198641015840 = 119864(2) 119909nearest larr Nearest(119866 119909)(3) 119909new larr steer(119909nearest 119909)(4) if Freeobsta(119909nearest 119909new)(5) 119875

1015840

= 1198751015840

cup 119909new(6) 119864

1015840

= 1198641015840

cup (119909new 119909nearest) (119909nearest 119909new)(7) 119883near larr Near(119866 119909new |119875

1015840

|)

(8) while 119909new isin 119883near(9) if Freeobsta(119909new 119909near)(10) 119864

1015840

= 1198641015840

cup (119909new 119909nearest) (119909nearest 119909new)(11) End(12) End(13) EndReturn 1198661015840 = (1198751015840 1198641015840)

Algorithm 2 Explore RRG (119866 119909)

Steer(119909 119910) Near(119866 119909rand |119875|) Freeobsta(119909 119910) are the same as RRGParent(119909near) returns the parent vertex of 119909near(1) 119875

1015840


1015840

= 1198751015840

cup 119909near(6) 119864

1015840

= 1198641015840

cup (119909new 119909nearest) (119909nearest 119909new)(7) 119883near larr Near(119866 119909new |119875|)(8) while all 119909new isin 119883near(9) if Freeobsta(119909new 119909near) and(10) cost(119909new) gt cost(119909near) + 119888(Dist(119909new 119909near))(11) 119909min = 119909new(12) End END(13) 1198641015840 = 1198641015840 cup 119909min 119909new

(14) while all 119909new isin 119883near119909min

(15) if Freeobsta(119909min 119909new) and(16) cost(119909near) gt cost(119909new) + 119888(Dist(119909new 119909near))(17) 119909parent larr Parent(119909near)(18) 119864

1015840

= 1198641015840

119909parent 119909near

(19) 1198641015840

= 1198641015840

cup 119909new 119909near

(20) End(21) EndReturn 1198661015840 = (1198751015840 1198641015840)

Algorithm 3 ExploreRRTlowast(119866 119909)

used for finite points in the Euclidean plane and now widelyused with a series of improved forms in the field of pathplanningVoronoi diagramgenerates topological connectionthe distances from the edges to the nearby obstacles are thesame

3D Voronoi first selects an initial site this sitersquos coor-dinates hold the property that the minimal distance to theobstacles nearby is the same Then it calculates new Voronoisites based on calculation of the Voronoi channel [88] anddefines the Voronoi net bounds The whole process stopswhen all the sites are recorded and all the channels are

obtained The calculation of the Voronoi channel is used tochoose a Voronoi site and the equation of Voronoi channelfor the triplet of objects 119894 119895 119896 is as follows

119889119894(119903) = 119889

119895(119903) = 119889

119896(119903) (1)

where 119889119894(119903) is the minimal distance from a give point (or

robot as amass point with 3D coordinate) to the surface of 119894thobstacle in 3D space Equation (1) extends the characteristicsof general Voronoi if there exists a small shift V along the


119880 is the vertices set according to the node choose method claimed aheadFreeobsta(119909rand 119906)means no obstacle between 119909rand and 119906119875 is the path vertex set 119864 is the path edges set(1) 119875

1015840

= ⊘ 1198641015840 = ⊘(2) for 119894 = 0 to 119899(3) 119909rand larr the 119894th sample state in 119883free(4) 119880 larr (119870 PRM or 120575-ball or 119870 cap 120575)119901(5) 119875 = 119875 cup 119909rand(6) for each 119906 isin 119880(7) if Freeobsta(119909rand 119906)(8) 119875 = 119875 cup (119909rand 119906) (119909rand 119906)(9) End End(10) EndReturn 119866 = (119875 119864)

Algorithm 4 Whole PRM

channel and the point 1199031015840 = 119903 + V should satisfy the constraintequation

119889119894(119903 + V) = 119889

119895(119903 + V) = 119889

119896(119903 + V) (2)

With respect to (1) linearization of this system should obey

(nabla119889119894sdot V)1003816100381610038161003816119903 = (nabla119889

119895sdot V)

10038161003816100381610038161003816119903= (nabla119889

119896sdot V)1003816100381610038161003816119903 (3)

This ensures finding the direction of the displacement V thuswe can get a new point 1199031015840 along the vector

An estimation function (4) is used to improve thecomputer-based accuracy that is controlling the deviation ofthe trajectory from the Voronoi channel

Φ = (119889119894minus 119889119895)2

= (119889119894minus 119889119896)2

= (119889119895minus 119889119896)2

(4)

Let Φ = 0 ensure that the point is on the Voronoichannel If Φ gt 120575

2 (120575 is desired value) then it returns tothe Voronoi channel by applying gradient decay procedureHowever in order to ensure fast convergence the preliminarysampling of each state is very important thus this kind ofalgorithm is classified as sampling based (even though itintroduced information adjustment feedback to correct theinitial sampling)

Voronoi generates a global graph or local graph butalmost the same as RRG and PRM it also cannot generatethe shortest path at the same time Thus it seeks help fromDijkstrarsquos algorithm Alowast Dlowast and so forth This paper definesthree steps to Voronoi path generationmethods (a) samplingthe environment or just seeking help fromother environmentconstruction methods (b) generating a 3D Voronoi graphand (c) employing a search algorithm to find theminimal costpath globally

Luchnikov et al [88] first proposed an elaborate 3DVoronoi diagram construction method which solves 3Dcomplex system path planning problem Reference [35ndash37]improved it to a further stage by proposing a radial edge likedata structure which is capable of dealing with topologicalcharacteristics of Euclidean Voronoi diagram of spheres Thetopological characteristics are region face edge vertex oop

and partial edge which contain the geometric informationand adjacency relationship When implemented in realityLifeng and Shuqing [38] introduced the method that usesgeographical information system to generate the environ-ment nodes and combined with Dijkstrarsquos algorithm to findthe shortest path Sharifi et al [97] assigned Voronoi regionto a group of UAV to solve the problem of coverage planningfor an environment Voronoi was combined with potentialfield method in [39 40] which is efficient to guaranteefast convergence Voronoi channel is built mainly based onstatic obstacles authors in [41] tackle dynamic threats byadding a bound to the Voronoi channels and the idea canbe introduced to 3D environment

44 Artificial Potential Algorithms Since first proposed byKhatib [86] potential field methods have been widelyresearched and implemented universally because of its lowcomputational complexity Potential field methods are basedon the idea of assigning a potential function (relationshipbetween obstacle free and obstacle space) to free space andsimulating the vehicle as a particle reacting to force dueto the potential field It computes the goal attraction andobstacle repulsion simultaneously and guides the robot alongthe total force gradientThe potential field can be representedas

119880119905(119883) = 119880

119886(119883) + 119880

119903(119883) (5)

where in formula (5) 119880119905(119883) is the total potential at state

119883 119880119886(119883) is the attractive potential at state 119883 and 119880

119903

denotes the repulsive potential at state 119883 Given nabla119880119903(119883) =

minussumall119891119903119894(119883) that is the virtual attractive force (from all

neighbor obstacles also please keep in mind the relationbetween the gradient of potential field and virtual force canbe negative or positive for your consideration) then the forcefield at state119883 can be represented as

119891119905(119883) = minusnabla119880

119905= 119891119886(119883) +

allsum

119894=1

119891119903119894(119883) (6)


For the attractive potential

119880119886= 119896119886

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817 (7)

119891119886(119883) = minusnabla119880

119886= minus119896119886

(119883119866minus 119883)

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817

(8)

where 119883119866

is the goal node 119883119866minus 119883 is the error vector with

respect to the goal node and 119891119886(119883) is the attractive force at

119883 Hence attractive force increases with the distance betweencurrent node and goal node For the repulsive potential

119880119903119894=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894)) if 120578

119894(119883) le 120578

0119894

(9)

119891119903119894(119883) = minusnabla119880

119903119894(119883)

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894

1205782

119894(119883)

(1

120578119894(119883)

minus1

1205780(119894))nabla120578119894(119883) if 120578

119894(119883) le 120578

0119894

(10)

where 120578119894(119883) is the distance from 119909 to obstacle region which

has been partitioned in convex components the distance canbe defined flexibly Δ120578

119894(119883) is a polar value 119891

119903119894is the 119894th

obstacle component repulsive force to 119883 and 1205780119894

is the safemargin of 119894th convex component which is defined by certainobstacle For field implementation APF tends to be morecomplicated as the bounder of the obstacles is not constantthus the time efficiency is constructed basedmainly on initialguess thus we classify it as sampling based algorithm

However such methods are incomplete because they areprone to drop into local minima area Many studies havebeen done to help potential field algorithms to overcomethe local minima by generating navigation functions orcomputing the potential with constraints Reference [42]proposed a harmonic potential method to solve the localminima problem by using Laplacersquos Equation to constrain thegeneration of a potential function Rimon and Koditschek[43] proposed a Morse function with a single minimum atthe desired destination strategy to form a strong stable robotnavigation method to jump out of the local minima and thisis the first formally proposed navigation method Authors in[44] imported the idea of Hamilton-Jacobi-Bellman to forma HJB function

min119906

⟨119881 (119909) 119906⟩ = minus1 (11)

where 119906 is the control and 119881(119909) is the potential factor Thisfunction generates a unique global minimum value thusit is able to yield a global feasible path that jumps out oflocal minima and [45 46] put it to a higher stage Ge andCui [47] solved the goals nonreachable with obstacle nearby(GNRON) problem by introducing a new repulsive

119880119903119894

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894))

2

120588119899

(119883119883119866) if 120578

119894(119883) le 120578

0119894

(12)

where 120588119899(119883119883119866) is theminimal distance between the robot119883

and the goal119883119866 120588119899(119883119883

119866) ensures the total potential119880

119886(119883)

arrives at its globalminimum that is 0 if and only if119883 = 119883119866

45 Analysis Sampling based algorithms exploring dependsmuch on initial guess and this guess repeats every step Thealgorithms may take advantage of nearby collision detectionor potential adjustment Although this leads to loss ofcompleteness and inexplicit construction of the environmentsometimes this can somehow reduce the dependence onenvironmentmodel and thus enable them to be implementedin various environments We summarize all the advantageand weakness of this kind of algorithms and analyze eachsubcategory in detail as illustrated in Table 2

Except for algorithms analyzed in Table 2 algorithmssuch as visibility graphs [18] and corridor map [98] alsobelong to sampling based algorithmsVisibility graphmethodlikes a simple version of obstacle-based PRM which isproposed by Amato et al [32] Corridor map method likesoctree-structure PRM introduced by Yan et al [6] Theyboth defined a cell decomposition method to constructthe workspace However such as PRM and Voronoi thesemethods need a node search algorithm to achieve the bestpath

5 Node Based Optimal Algorithms

Node based optimal algorithms are classified by the reasonthat they deal with nodesrsquo and arcsrsquo weight information(but not limited to these or sometimes called grid) theycalculate the cost by exploring through the nodes thus tofind the optimal path This kind of algorithms is also callednetwork algorithms [99] which means they search throughthe generated network and it means the same with nodebased algorithms

51 Dijkstrarsquos Algorithm Named after Dijkstra [21] Dijkstrarsquosalgorithm targets for finding the shortest path in a graphwhere edgesrsquoarcsrsquo weights are already known Dijkstrarsquosalgorithm is a special form of dynamic programming and it isalso a breath first search method It finds shortest path whichdepends purely on local path cost When applying 3D spacea 3D weighted graph must be built first then it searches thewhole graph to find theminimum cost path A generalizationof Dijkstrarsquos algorithm is shown below with pseudocode inAlgorithm 5

Authors in [100] proposed an improved Dijkstrarsquos algo-rithm by adding a center constraint it works well in tubularobjects which is first proposed by [101] The work [102]showed that 3D GIS environment combined method maybe a practical way to implement in real outside world andexperimental results are provided to prove that Dijkstrarsquosalgorithm acts well enough But Dijkstrarsquos algorithm reliesmuch on the priority queue 119876 data structure type whichinfluences the total exploring time

52 A-Star (119860lowast) A-star (Alowast) [22] is an extension of Dijkstrarsquosalgorithm which reduces the total number of states byintroducing a heuristic estimation of the cost from the


Table 2 Analysis of sampling based algorithms

Method type Shortcoming AdvantagesShortcomings Improvement

RRTSingle path [28 29]

Low time complexity fastsearching abilityNonoptimal [29 30]

Static threat only [31]

PRMExpensive collision check [6 32 33] Appropriate for complex

environments and replanningsituations

Static threat only [34]Nonoptimal [30]

VoronoiIncomplete representation [35ndash38]

Easy to implement on-line andignoring collision checkingNonconvergence [39 40]

Static threat only [41]Artificial potential Local minima [42ndash47] Fast convergence

119876 is the priority queue 119909119866is the goal

119897(119909 119906) return the cost to apply action 119906 from 1199091198621015840 is the best cost-to-come known so far

119880119909is the finite action space

1199091015840

= 119891(119909 119906) isin 119883 is a state transition function(1) 119862(119909) = infin | (119909 = 119909init) 119862(119909init) = 0(2) 119876Insert(119909

119868)

(3) while 119876 not empty(4) 119909 larr 119876GetFirst()(5) for all 119906 isin 119880(119909)(6) 119909

1015840

larr 119891(119909 119906)

(7) if 119862(119909) + 119897(119909 119906) lt min119862(1199091015840) 119862(119909119866)

(8) 1198621015840

(119909) = 119862(119909) + 119897(119909 119906)

(9) if 1199091015840 = 119909119866

(10) 119876Insert(1199091015840)(11) END END(12) END END

Algorithm 5 Dijkstrarsquos algorithm

current state to the goal state The heuristic function can bedesigned to obtain the constraints while the estimation mustnever overestimate the actual cost to get the nearest goal nodeBy applying this guiding like heuristic Alowast can converge veryfast and ensures optimality as well

Alowast is proposed by introducing an evaluation function(13) which consists of postcalculation toward the initial stateand heuristic estimation toward the goal

119891 (119899) = 119892 (119909) + ℎ (119909) (13)

where 119892(119909) is the cost from initial state 119909init to current state119909 which is the same as 119888(119909) in Algorithm 5 ℎ(119909) is theheuristic estimation of the cost of an optimal path fromcurrent state 119909 to goal state The estimation ℎ(119909) of each statetends to be close to the real cost thus Alowast has a faster speedto converge based on comparison of the cost of neighborsComparedwithDijkstrarsquos algorithmAlowast obtains a faster speedto converge

For 3D environment Alowast has been widely implementedAmato et al [32] constructed the feasible road map by

applying PRM and then adopted Alowast to execute best routeexploration Authors in [6] implemented Alowast with UAVplatform with an octree based PRM Niu and Zhuo [53]introduced ldquocellrdquo and ldquoregionrdquo conception to enhance theenvironment understanding of Alowast thus enabling flexible rep-resentation of 3D environments Koenig and Likhachev pro-posed an environment-representation varying adaptive Alowastthat is Lifelong Planning Alowast (LPA) [48] LPAlowast can adapt toenvironment changes by using previous information as wellas iterative replanningWilliams andRagno [49] propoundeda conflict-direct Alowast it accelerates the exploring process byeliminating subspaces around each state that are inconsistentIt is proposed in [50] that Alowast can choose any state to beparent state thus resulting in amore flat turning angle namedThetalowast The algorithm has the ability to be able to obtainsystem constraints thus it can find shorter and more realisticpath De Filippis et al [103] implemented both Thetalowast andAlowast in 3D environment and an experimental comparison isgiven to prove that Thetalowast reduces the searching comparedto Alowast Although Thetalowast acts well compared to Alowast but whenapplied to 3D environment it consumes much time to checkunexpected neighbors Line-of-sight check method calledlazy Thetalowast [51] is proposed to avoid unnecessary check ofunexpected neighbors Authors in [54] introduced a methodto reuse information from previous explorations and updateinformation through the affected and relevant portions ofthe exploring space This may cause extra computationalconsumption but it can tackle dynamical threat and convergefast

53 D-Star (119863lowast) D-star (Dlowast) [23 52] short for dynamicAlowast is famous for its wide use in the DARPA unmannedground vehicle programs Dlowast is a sensor based algorithm thatdeals with dynamic obstacles by real time changing its edgersquosweights to form a temporal map and then moves the robotfrom its current location to the goal location in the shortestunblocked path

Dlowast As well as Alowast evaluates the cost by consideringthe postcalculation and forward estimation Dlowast maintains alist of states which is used to propagate information aboutchanges of the arcs cost function The evaluation function isrepresented as


Table 3 Analysis of node based optimal algorithms


Dijkstrarsquos algorithm High time complexity [22 23 48ndash52] Easy to implement for various environmentsStatic threat only [23 52]

AlowastHeavy time burden [48 50 51]

Fast searching ability enabling implementation on-lineNonsmoothness [50 53]Static threat only [23 52 54]

Dlowast Unrealistic distance [55 56] Fast searching ability dealing with dynamic environments

1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)

However differing from Alowast ℎ1015840(119909) is not necessarilythe shortest path length to the goal compared to Alowast alsocomputation of ℎ

1015840

(119909) assumes that the robot can passthrough obstacles At each time it updates a minimumheuristic function when it encounters new obstacles and thewhole graph thus enabling efficient searching in dynamicenvironments

Smirnov [55] considered the worst travel distance caseof Dlowast and restrained its lower bound as Ω((log |119881|log log |119881|)|119881|) steps and upper bound as119874(|119881|2)The boundis used to solve the problem that Dlowast uses unrealistic distancein its graph and there is a large gap between the lower boundsand upper bounds Based on [55] Tovey et al [104] put itforward by adding more tighter bounds for Dlowast Koenig andLikhachev [56] extended LPAlowast to the case where the goalchanges between replanning episodes it is like a simplifiedversion of Dlowast called Dlowast Lite but it is proposed based onLPAlowast

54 Analysis Table 3 provides a straightforward summa-rization of node based optimal algorithms As the nameimplies node based optimal algorithms deal with node andarc information However as the nodes and arcs provide anincomplete structure of the configuration space this kindof method can only achieve the best result limited by therepresentation of environment This kind of algorithms issingle search methods It cannot generate multipath for fleetFor real time implementation Dijkstrarsquos algorithmrsquos timecomplexity is 119874(1198992) (119899 is the number of the nodes) Alowast andDlowast reduce the complexity to a lower degree which enableson-line implementation

6 Mathematic Model Based Algorithms

Node based optimal algorithms use grids to represent con-figuration space meanwhile this kind of methods assumesthe robot as a point and only considers the accelerationand velocity constraints Thus it is not complete to representthe environments as well as consider system dynamicsMathematic model based algorithms optimize by describ-ing kinodynamic constraints in combination of polynomialforms They can model the environment as a time variantsystem (sometimes even a model and time variant system)which is synchronization to the current location of the robot

Due to the fact that this kind of algorithms can han-dle dynamics constraints to achieve cost optimum someresearchers regard mathematic optimization methods astrajectory planning method However mathematic modelbased algorithms are proposed to solve the problemof findinga path or trajectory locally or globally [45 105] In order todistinguish from trajectory planning methods we provide astronger definition compared to Definition 4

Definition 5 Trajectory planning assumes output to be timecontinuous and must be able to ensure control limitationwhich is still a part of the whole path

According to Definition 5 above local mathematic opti-mization methods also partly belong to path planning fieldAuthors in [105] transformed kinodynamic constraints intoa bound of linear or nonlinear constraints and then usedmathematic programming methods to find the boundedoptimal path Miller et al [45] modeled the path planningproblem in an optimal control form which combines costcriterion and Hamiltonian function to form a boundaryvalue problem (BVP) to find a realistic optimal path globallyChamseddine et al [63] introduced flatness based method tolinearize the nonlinear bounds into polynomial form and abang-bang control is employed to generate a global optimalpath Other mathematic optimization methods such as levelset method and support vector machine method also belongto this kind

61 Linear Algorithms For linear algorithm form based pathplanning problem this kind of algorithm tends to have thefollowing form

119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)

where 119869cost-goal is a cost function that takes into kinodynamicconstraints as as well as the expected properties such as min-imum distance energy and threat 119891(119906 119909 119910 119911) represents


Maximumprinciple

resultsin optimal path

Form cost functionHamiltonian function

Min goalconditions

Kinodynamicconstraints

Figure 11 Basic optimization problem steps

the control factor 120593(119909 119910 119911) is the kinodynamic constraintwhich acts as penalty function and 119877(119909 119910 119911) is the exploringstep reachable region function to ensure a strong convergenceability to the goal Equation (16) is the initial condition (17) isthe final condition (18) is algebraic path constraints and (19)is the control constraints

Linear algorithms have the ability to describe the environ-ment completely meanwhile they can model the kinematicsand dynamics constraints Furthermore linear algorithmscan handle control disturbance or model uncertainty Mixed-Integer Linear Programming (MILP) methods combinebinary and integer logical constraints and are most com-monly used because they closely represent the environmentand system Reference [3 5 106] implemented MILP respec-tively in aerial underwater and ground robots Bhattacharya[64] directly supported a free MATLAB toolbox which iscalled OPTRAGEN to solve the MILP problem Masehianand Habibi [62] used 0-1 binary integer to represent the pathlength operator and then solved the path planning problemusing binary integer programming

62 Optimal Control Path planning problem can be posedin an optimal control form where optimal control canfind the state and control oriented path based on a setof differential equations [107] Optimal control can beinterpreted as an extension of the linear algorithms to aninfinite number of variablesrsquo condition with the ability tomodel uncertainty as linear chance constraints much moreeasier

A basic optimization problem is depicted in a flowchartform in Figure 11 where the initial state (or current state)and goal state are included in the constraints to ensurecompleteness

For optimal control case a general system model isconsidered in time continuous form

= 119891 (119909 (119905) 119906 (119905)) (20)

where 119909(119905) represent the evolving state and 119906(119905) is the controlparameters The minimizing criterion is

119871 = 120601 [119909 (119905) 119906 (119905)] (21)

With the necessary constraints

120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)

Then cost function (an recommended performanceindex) can be derived by including all constraints above

119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)

Then we can achieve the Hamiltonian

119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)

where (22) is the initial condition and (23) is the finalcondition Hamiltonian is used to solve the optimal problembased on the maximum principle and then follow the typicaloptimal solving procedure to generate an optimal pathglobally This paper only explains in a general form eachequation can be extended to contain much more constraints

Tisdale et al [58] implemented discrete receding-horizoncontrol (RHC) in UAV which is a variant of optimal controlChen and Schwartz [65] described the optimal controlproblem in detail and proposed a free MATLAB toolboxRIOTS 95 and in [108] they further implemented it to solvemodel predictive problem

63 Analysis This kind of methods maintains a completeform to describe the states and environmental variables andwhen applied in 3D cluttered environments these methodscan adapt themselves by employing much more constraintsaccording to the environments These algorithms tend tohave a much more complex formulation that is a heavycomputational burden There exists a method that can solvethe weakness that is discrete decisions in an optimizationprocedure The method allows problems to be solved on-lineflexibly by devising the environment [58ndash60] For exampleBellinghamet al [61] only usedMILP to justify the local statesand thus increase the whole exploring process a lot Based onthe analysis above this paper provides a summarization inTable 4


Table 4 Analysis of mathematic model based algorithms


Mathematic model based algorithms High time complexity [58ndash62] Containing almost all the information to generatean optimal path indeedNo analytic solutions [63]

Free tool OPTRAGEN [64] RIOTS 95 [65]DIDO [66] SeDuMi [67] and so forth

7 Bioinspired Algorithms

Path planning is attributed to the top layer of a robot controlprocess which enables robots to work without (or withlittle) help from man The motivation of planning a pathin real cluttered environments is that the robots shouldobtain the ability to accomplish the mission by itself withoutsupervision For bioinspired algorithms they originate fromimitating the way how humans or other natural creaturesbehave or think and they form a family of a series ofalgorithms which can solve NP-hard problems to generate anear optimal path

There are two subcategories of bioinspired algorithmsone is Evolutionary Algorithm which stems from analyzingthe behavior of a certain species another is Neural Networkalgorithmwhich follows the way how inner neuron processesthe information They belong to different level accordinglyand this paper discusses them respectively For EvolutionaryAlgorithms they work almost with the same mechanismthus this paper mainly analyzes two relative popular algo-rithms Genetic Algorithm and Ant Colony OptimizationAlgorithm

71 Evolutionary Algorithm Evolutionary Algorithm is anumbrella name which includes Genetic Algorithm (GA)Memetic Algorithm (MA) Particle Swarm Optimization(PSO) Ant Colony Optimization (ACO) and Shuffled FrogLeaping Algorithm (SFLA) EA was propounded to solvethe problem where traditional linear programming anddynamic programming often fail to solve NP-hard problemswith large number of variables EA is a stochastic searchapproach that imitates the metaphor of natural biologicalevolution and social behavior The firstly proposed andnow widely implemented Evolutionary method is GA theninspired by different natural process four other methodsdeveloped

Figure 12 illustrates a typical flowchart of EvolutionaryAlgorithm which is proposed by Dong and Juris [109]Evolutionary Algorithms start by selecting randomly feasiblesolutions as the first generation Then it takes the envi-ronment robotrsquos capacity (dynamic ability) goal and otherconstraints into consideration to evaluate the fitness of eachindividual In the next step a set of individuals is selectedas parents for the next generation according to their fitnessThe last step is a mutation and crossover step The wholeprocess is performed in an iterative way and stops the processwhen a preset goal is achievedThe best fitness individuals aredecoded as the optimal path nodes

711 Genetic Algorithms Holland [89] firstly introducedGA and now it is the most popular population-basedoptimization method The basic version of GA defines acost function to evaluate the potential solutions Then apartly random crossover operator takes two parents fromthe population set and recombines them in some way Themutation operator tries to modify the solutions and aims toachieve a valid solution in order to escape local optimality

GA holds the process that all individuals can exchangeinformation by doing this way this nest generation canconverge very fast by using this information However ifthe population becomes too similar and loses populationdiversity it often leads to premature convergence If thereexists too much population diversity it may result in a heavycomputational burden to investigate the poor solutionsGA needs to repeatedly evaluate the fitness of the currentgeneration which also causes a high computational burden

Fonlupt et al [70] proposed a cooperatingGA to solve thepremature convergence problem of typical GA The methodholds the concept that when a certain GA is stuck in localminimum then another GA might provide a feasible pathpoint to allow it to search again Hacioglu and Ozkol [71] andHacioglu and Ozkol [72] solved the problem by introducingan idea that is periodically applying a vibrational mutationoperator to the whole population Therefore it becomespossible to escape from local optimums and then to obtain aglobal optimal path To avoid time consumption with a highburden authors in [68] proposed a combination of GA andVoronoi method The Voronoi diagrams are constructed byusing fuzzy 119888-means clustering method to generate the firstgeneration which accelerates the convergence speed RRTobtains the merits of random exploring which can enableescaping of local optimum Authors in [69] employed RRTto generate the first generation of chromosomes to achieveglobal optimality

712 Ant Colony Optimization Animals such as ants couldmanage to establish shortest path from their colony tothe feeding source and back home by group cooperationresearchers mimic the behavior and proposed Ant ColonyOptimization (ACO) method ACO introduces two basicconcepts which are ldquointensity of trailrdquo and ldquovisibilityrdquo to formthe transition probability which at last decides which way togo thus to formulate the shortest path

ldquoIntensity of trailrdquo on edge (119894 119895) at time 119905 + 119899 is expressedby the following formula

119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)


Evaluate fitnessof currentgeneration

Initialization

Selectparents

Decode Path

Mutationcrossover

Figure 12 Evolution process for path planning

where 119879119894119895(119905 + 119899) represents the information about how many

ants in the past have chosen the same edge (119894 119895) and 120588 is aweight to represent how much information is left between 119905and 119905 + 119899

Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896

if 119896th ant uses the edge (119894 119895)

0 if otherwise

(27)

where Δ119879119894119895

is the sum of all 119898 antsrsquo ldquopheromonerdquo laid onthe edge (119894 119895) between time 119905 and 119905 + 119899 Δ119879119896

119894119895is the 119896th ant

ldquopheromonerdquo laid 119876 is a constant and 119871119896is the 119896th ant tour

lengthldquoVisibilityrdquo on edge (119894 119895) can be described as

120578119894119895=

1

119889119894119895

(28)

where 119889119894119895

is the Euclidean distance between 119894 and 119895 and 120578119894119895

determines the degree of how close the state 119894 is to state 119895ACO aims to find the best path by evaluating the

pheromone density in each step The process runs flexi-bly in dynamic environments only needing to change therepresentation of ldquointensity of trailrdquo of a certain edge Thealgorithm is proved to be able to deal withmultiobjective pathplanning problems and it is also able to tackle continuousplanning problems [110] ACO is now widely implementedin 3D environment [7 73] for path planning However itmust be emphasized that the basic ACO is not applicableto handle vast size of pheromone matrix in practical timewith computer memory limitation Thus it is validated insimulation but not practical in real time planning situationsin most cases

Authors in [7] proposed a differential evolution (DE)ACO method by applying differential evolution to opti-mize the pheromone tail DE is a simple population-basedalgorithm it tends to be more efficiency thus the worktakes advantage of DE crossover operation to achieve fasterconvergence Method proposed in [73] started with relative

coordinates which can avoid rotation transformation forUAV to reduce time consumption for generating an optimalpath Saber and Alshareef [74] came up with using Alowast toincrease the local searching ability and introduced proba-bilistic nearest neighbor method to estimate the pheromoneintensity it is proved to be effective

The other three Evolutionary Algorithms Memetic Algo-rithm [90] Particle Swarm Optimization [91] and ShuffledFrog Leaping Algorithm [93] all almost share the sameexploring process and their shortcomings and advantages arealso almost the same Although compared to GA and ACOthey have a lower degree of implementation still a lot ofworks[4 111 112] have been done with these methods

72 Neural Network Another subcategory of bioinspiredpath planning method is Neural Network (NN) NN wasintroduced first introduced by Glasius et al [113] to avoidobstacle as well as navigation and then became popular andimplemented to path planning in various areas [114ndash116]NN approach aims to generate a dynamic landscape in aneural-like form It shares some merits as Artificial PotentialField method the unsearched areas attract the robot in theentire space globally A typical shunting equation is used toexpress the dynamics of a robot in neuron network which isrepresented in the following form

119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)

where 119909119894denotes the neuron activity of the 119894th neuron

Parameters119860 119861 and119863 are nonnegative constants represent-ing the passive decay rate and the upper and lower boundsof the neural activity respectively [119868

119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and

[119868119894]minus are the excitatory and inhibitory inputs 119868

119894is the external

input to the 119894th neuron and it is defined as 119868 = 119864 if it is a safeunexplored area 119868 = minus119864 if it is an obstacle area 119868 = 0meansother cases Here 119864 ≫ 119861 is a very large positive constant

For NN in each step it first chooses the maximal neuralactivity among the neighbor neurons then the next location


Table 5 Analysis of bioinspired algorithms


GA High time complexity [68 69] Able to solve NP-hard andmultiobjectives problemsPremature convergence [70ndash72]

ACO High time complexity [7 73 74] Able to deal with multiobjectives andcontinuous planning problems

PSOHigh time complexity

[75] It acts faster than GA and can deal with alow number of individuals problemsPremature convergence

Parameter sensitive [76]

SFLA High time complexity [77] It is more efficient than PSO and canachieve global convergence fasterParameter sensitive [78]

MA High time complexity [79]It is more efficient than GA in pathsmoothness and with low computationalcomplexity

NN High time complexity [80ndash82] Stable under sudden changes in thenetworkRelying on suitable rules and organisms [83]

is determined by this maximal neural activity Equation (29)guarantees that the positive neural activity can propagateto all the free unexplored space with the maximal neuralactivity but negative activity only stays locally Although NNattracted so much attention it shares the weakness as otherbioinspired algorithms that it cannot form canonical rulesand model thus the reliability and time consumption arenot guaranteed Even though Hopfield model is introducedto solve the weakness [83] this is only applicable to certainproblems

Kassim and Kumar [80] propounded a new ldquowaveexpansion neural network (WENN)rdquo which introduces gridpotential for path planning the method was implemented to3D workspace and proved to have time efficiency RecentlyKroumov et al [81 82] improved themethod by combinationwith potential field methods this method accelerates thespeed of basic NN But according to [80] it clearly can beseen that when extended to 3D environments the numberof neighbor neurons will increase to 26 which means thecomputation complexity will explode simultaneously Thus itis applicable to implement real time

73 Analysis Bioinspired algorithms stem from mimickingthe natural behavior they specify a set of optimizing rulesas well as the model of processing the information Then thewhole process executes based on the rules and models byiterative optimization Thus the final results rely much onthe rules and model proposed and the time consumptioncannot be guaranteed Often if the environment becomesmore complex they need a large computational resource tofind a solution to meet the expectation (Table 5)

EA is named as a kind of population-based algorithm allof its subcategories share almost the same procedures repro-duction mutation recombination and selection Althoughthis kind of algorithms tends to be time consuming andpremature convergence it can deal with multiobjectivesproblems and solve NP-hard problems GA is the mostpopular Evolutionary Algorithm and PSO is proved to act

much faster than GA but PSO is sensitive to parametersPSO can be solved by hybrid approach [76] but also holdsthe same shortcoming as GA ACO can find optimal pathwithout premature convergence but it is impossible for thebasic ACO to deal with real complex environment SFLAoperates the local exploration by using PSO-like methodbut it shuffles virtual frogs periodically with ensuring globalexploration The PSO-like method relies strongly on suitableparameters [78] MA is sometimes called hybrid GA whichdiffers from GA in that MA shares chunk of the gene ratherthan a neat crossover in chromosomes Since MA coupledwith a learning procedure to perform local refinements it stillhas a more heavy computational burden NN is not sensitiveto environment changes but it relies too much on its model

8 Multifusion Based Algorithms

Multifusion based algorithms manage with problems whereusually a single approach proposed cannot work to findan optimal path individually When faced with unknownenvironments with whether dynamic threat or static threatnot all traditional single planning approaches can fulfill thetask of obtaining cost minimum or fast convergence orcomputational efficiency For example PRM cannot generatean optimal path itself which is the same as Voronoi Potentialfield methods often jump into local minimum node basedoptimal algorithms need preknown environmental skeletoninformation Mathematic model based algorithms tend tobe time consuming and unable to solve NP-hard problemwith varying environments bioinspired algorithms vary theirperformance with the model diagram and have heavy timecomplexity Thus researchers try to introduce a combinationof different approaches to form a fast searching and globaloptimal algorithm

Yan et al [6] used 3D grid to represent the environmentand 3D PRM to form a road map in obstacle free space atlast Alowast optimal search algorithm combined to achieve anoptimal path Masehian and Amin-Naseri [39] introduced


a visibility graph Voronoi diagram and potential field (VVP)integrated algorithm If the extension of VVP algorithm to3D space is investigated it shows effective tradeoff betweenthe shortest and safest path Schoslashler et al [85] combinedvisibility graph and Dijkstrarsquos algorithm (or Geodesics) tofind an optimal solution for path planning problem in 3DJaishankar and Pralhad [84] planned outdoor environmentpath by citing GIS-MCDA approach which first synthesizesa variety of information to generate a ldquocombined gray levelimagerdquo then an optimal searching algorithm asAlowast Dlowast or EAis used to achieve an optimal path A hybrid of mathematicmodel based algorithms with EA [117] is proposed to solveNP-hard problem where general EA tend to be prematureA lot of works adopt this idea this paper gives a pioneeringwork of coming up with a taxonomy of this kind of methods

Based on the principle of each algorithm this paper clas-sifies all of these multifusion based algorithms into two cat-egories (a) Embedded Multifusion Algorithms (EMA) and(b) Ranked Multifusion Algorithms (RMA) EMA combinesseveral algorithmsrsquo advantages together these algorithmswork in simultaneous model thus ensuring a tightly cou-pled approach to achieve better performance RMA meanseach algorithm works in separate level where they workrankly Table 6 illustrates several typical algorithms of eachcategory

9 Analysis and Conclusion

This paper analyzes a certain aspect of providing a compre-hensive knowledge of 3D path planning including canonicaldefinition and basic knowledge of each kind of algorithmsas well as applicable area By analyzing all these algorithmswhich have already proved to be successful this paperclassifies all the approaches into five categories samplingbased algorithms node based optimal algorithms mathe-matic model based algorithms bioinspired algorithms andmultifusion based algorithms This paper first lists the ele-ments of each category and then supports a critical discussionof each algorithm During each discussion process this paperanswers four important questions respectively (a)what is thetaxonomy of these 3D path planning algorithms (b) How dothese algorithms work in order to find the path (c) Why is itsuitable or not suitable for such environment

According to the analysis we supported several conclu-sions can be drawn as illustrated in Table 7 with the detailsof time complexity static (S) or dynamic (D) environmentalapplicability and real time applicability

(1) Sampling based algorithms all share themerits of usingan initial guess they differ in how to do postprocessing toensure completeness or optimum The random initial guessensures escaping of local minimum and this kind of algo-rithms does not rely much on environmental representationThis kind of algorithms can be further classified as active andpassive where active algorithm can find the optimal path bytheir own but passive algorithms cannot These approachesare appropriate for on-line implementation as they have ahigh time efficiency with the ability to handle dynamic andstatic threats

Table 6 Typical elements of multifusion based algorithms

Name ofsubcategory Typical elements of each subcategory

Embeddedmultifusionalgorithms

RDR [34] VVP [39]Voronoi potential field algorithms [39 40]neural network potential field algorithms[80ndash82]hybrid bioinspired [7 74 76 78] and so forth

Rankedmultifusionalgorithms

PRM node based optimal algorithms [6]GIS-MCDA algorithms [84]visibility graph node based optimal algorithms[85]visibility graph geodesics algorithm [85]sampling based EA [68 69] and so forth

(2) Node based optimal algorithms are grid based explor-ing algorithms they commonly tackle with nodearc infor-mation which can transform distance between nodesarcsinto weight They originate from dynamic programmingapproaches and thus cannot further optimize the resultbeyond the decomposition of the environment The resultsof this kind of algorithms rely much on the preconstructedgraph and can be combined with other methods to achieveglobal optimal

(3) Mathematic model based algorithms aim to describethe whole workspace in a mathematic form with the advan-tage of describing all constraints with differential equationsThey can easily represent dynamic constraints and kinematicconstraints This kind of algorithms gives an overall consid-eration to safety reliability and efficiency and then boundsthe cost function tightly Although this kind ofmethods loadsa heavy computational burden on computer it can performwell enough with the improvement of computer technology

(4) Bioinspired algorithms import heuristic idea and canexcellently deal with complex and dynamic unstructuredconstraints as well as NP-hard problem This kind of algo-rithms optimizes the path by mutation (iterative optimiza-tion) but the process also brings problem simultaneouslyThe mutation process repeats until final goal is achievedalmost all bioinspired algorithms have the shortcoming ofhaving a long dealing period Thus this kind of algorithmsis suggested to work off-line even though they can handledynamic threats

(5) Multifusion based algorithms synthesize several algo-rithmsrsquo advantages together to achieve global optimal andcost minimum This kind of algorithms imports the ideaof complementation that is merging the merits of severalalgorithms These algorithms have ability to achieve severalgoal simultaneously thus it often happens that sometimesseveral simple relativemethods combine to form a rather wellperformedmethodThese methods are designed to work realtime with strong environmental adaption

3D path planning approaches explode these years butproblems such as real time planning complete informationexpressing and complex environments modeling still havenot been completely solved yet According to Section 8


Table 7 Properties of each kind of methods

Method Time complexity SD environment Real timeSampling based algorithms 119874(119899 log 119899) le 119879 le 119874(119899

2

) S and (Part) D On-lineNode based algorithms 119874(119898 log 119899) le 119879 le 119874(119899

2

) S and (Part) D On-lineMathematic model based algorithms Depending on the polynomial equation S and D Off-lineBioinspired algorithms 119874(119899

2

) le 119879 S and (Part) D Off-lineMultifusion based algorithms 119874(119899 log 119899) le 119879 Depending on the algorithms On-line

this paper recommends a fusion of multipath planning andenvironment modeling methods this may become a keydirection for 3D path planning under complex situations andalso control uncertainty is recommended to be included

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is partially supported by National Science andTechnology Support under Grant no 61503369 and Grantno 61528303 which are highly acknowledged High acknowl-edgment goes also to Professor D Theilliol [118] who reallyprovided so much guiding of how to construct a nice workduring his visiting time

References

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[2] iRobot INC ldquoVacuum cleaning robotrdquo httpwwwirobotcomFor-the-HomeVacuum-CleaningRoombaaspx

[3] N K Yilmaz C Evangelinos P F J Lermusiaux and NM Patrikalakis ldquoPath planning of autonomous underwatervehicles for adaptive sampling using mixed integer linearprogrammingrdquo IEEE Journal of Oceanic Engineering vol 33 no4 pp 522ndash537 2008

[4] M P Aghababa ldquo3D path planning for underwater vehiclesusing five evolutionary optimization algorithms avoiding staticand energetic obstaclesrdquo Applied Ocean Research vol 38 pp48ndash62 2012

[5] R Yue J Xiao S L Joseph and S Wang ldquoModeling and pathplanning of the city-climber robot part II 3D path planningusing mixed integer linear programmingrdquo in Proceedings ofthe IEEE International Conference on Robotics and Biomimetics(ROBIO rsquo09) vol 6 pp 2391ndash2396 Guilin China December2009

[6] F Yan Y-S Liu and J-Z Xiao ldquoPath planning in complex 3Denvironments using a probabilistic roadmap methodrdquo Interna-tional Journal of Automation and Computing vol 10 no 6 pp525ndash533 2013

[7] H Duan Y Yu X Zhang and S Shao ldquoThree-dimensionpath planning for UCAV using hybrid meta-heuristic ACO-DEalgorithmrdquo Simulation Modelling Practice and Theory vol 18no 8 pp 1104ndash1115 2010

[8] F Schler 3d path planning for autonomous aerial vehicles inconstrained spaces [PhD thesis] Department of Electronic Sys-tems Faculty of Engineering and Science Aalborg UniversityAalborg Denmark 2012

[9] H M Choset Principles of Robot Motion Theory Algorithmsand Implementations MIT Press 2005

[10] SM LaVallePlanningAlgorithms CambridgeUniversity Press2006

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[12] Y B Sebbane LighterThan Air Robots Guidance and Control ofAutonomous Airships vol 58 Springer 2012

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[15] F Yan Y Zhuang and J Xiao ldquo3D PRM based real-time pathplanning for UAV in complex environmentrdquo in Proceedings ofthe IEEE International Conference on Robotics and Biomimetics(ROBIO rsquo12) vol 6 pp 1135ndash1140 Guangzhou China Decem-ber 2012

[16] T Leenknegt Three-Dimensional Path Planning in ComplexEnvironments Vakgroep Elektronica en InformatiesystemenUniversiteit Gent 2013

[17] A Mustard ldquoUnderwater environmentrdquo httpitsohucom20110512n280555394shtml

[18] S Flemming C-H Anders la and B Morten Configura-tion Space and Visibility Graph Generation from GeometricWorkspaces for UAVs Book Section 4 American Institute ofAeronautics and Astronautics 2011

[19] S M LaValle ldquoRapidly-exploring random trees a new tool forpath planningrdquo Tech Rep 98-11 Computer Science Depart-ment Iowa State University Ames Iowa USA

[20] L E Kavraki P Svestka J C Latombe and M H OvermarsldquoProbabilistic roadmaps for path planning in high dimen-sional configuration spacesrdquo IEEE Transactions on Robotics andAutomation vol 12 no 4 pp 566ndash580 1996

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[24] M Zefran V Kumar and C B Croke ldquoOn the generation ofsmooth three-dimensional rigid body motionsrdquo IEEE Transac-tions on Robotics and Automation vol 14 no 4 pp 576ndash5891998


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[26] A Majumdar A A Ahmadi and R Tedrake ldquoControl designalong trajectories with sums of squares programmingrdquo inProceedings of the IEEE International Conference on Roboticsand Automation (ICRA rsquo13) pp 4054ndash4061 IEEE KarlsruheGermany May 2013

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[29] R Kala ldquoRapidly exploring random graphs motion planning ofmultiple mobile robotsrdquo Advanced Robotics vol 27 no 14 pp1113ndash1122 2013

[30] S Karaman and E Frazzoli ldquoOptimal kinodynamic motionplanning using incremental sampling-based methodsrdquo in Pro-ceedings of the 49th IEEE Conference on Decision and Control(CDC rsquo10) pp 7681ndash7687 Atlanta Ga USA December 2010

[31] A Yershova L Jaillet T Simeon and S M LaValle ldquoDynamic-domain RRTs efficient exploration by controlling the samplingdomainrdquo in Proceedings of the IEEE International Conferenceon Robotics and Automation pp 3856ndash3861 Barcelona SpainApril 2005

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[38] L Lifeng and Z Shuqing ldquoVoronoi diagram and gis-based3d path planningrdquo in Proceedings of the 17th InternationalConference on Geoinformatics vol 5 pp 1ndash5 Fairfax Va USA2009

[39] E Masehian and M R Amin-Naseri ldquoA voronoi diagram-visibility graph-potencial field compound algorith for robotpath planningrdquo Journal of Robotic Systems vol 21 no 6 pp 275ndash300 2004

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[45] B Miller K Stepanyan A Miller and M Andreev ldquo3Dpath planning in a threat environmentrdquo in Proceedings of the50th IEEE Conference on Decision and Control and EuropeanControl Conference (CDC-ECC rsquo11) vol 6 pp 6864ndash6869 IEEEOrlando Fla USA December 2011

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[48] S Koenig and M Likhachev ldquoImproved fast replanning forrobot navigation in unknown terrainrdquo inProceedings of the IEEEInternational Conference on Robotics and Automation (ICRArsquo02) vol 1 pp 968ndash975 IEEE Washington Wash USA 2002

[49] B C Williams and R J Ragno ldquoConflict-directed Alowast andits role in model-based embedded systemsrdquo Discrete AppliedMathematics vol 155 no 12 pp 1562ndash1595 2007

[50] A Nash K Daniel S Koenig and A Felner ldquoThetalowast Any-angle path planning on gridsrdquo in Proceedings of the NationalConference on Artificial Intelligence vol 22 pp 1177ndash1198 AAAIPress MIT Press Vancouver Canada 2007

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[56] S Koenig and M Likhachev ldquoFast replanning for navigation inunknown terrainrdquo IEEE Transactions on Robotics vol 21 no 3pp 354ndash363 2005

[57] S Karaman and E Frazzoli ldquoSampling-based algorithms foroptimal motion planningrdquo International Journal of RoboticsResearch vol 30 no 7 pp 846ndash894 2011


[58] J Tisdale ZW Kim and J K Hedrick ldquoAutonomousUAVpathplanning and estimation an online path planning frameworkfor cooperative search and localizationrdquo IEEE Robotics andAutomation Magazine vol 16 no 2 pp 35ndash42 2009

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[60] C S Ma and R H Miller ldquoMilp optimal path planning forreal-time applicationsrdquo in Proceedings of the American ControlConference p 6 Minneapolis Minn USA 2006

[61] J Bellingham M Tillerson A Richards and J P How ldquoMulti-task allocation and path planning for cooperating UAVsrdquo inCooperative Control Models Applications and Algorithms pp23ndash41 Springer 2003

[62] E Masehian and G Habibi ldquoRobot path planning in 3D spaceusing binary integer programmingrdquo International Journal ofMechanical System Science and Engineering vol 23 pp 26ndash312007

[63] A Chamseddine Y Zhang C A Rabbath C Join and DTheilliol ldquoFlatness-based trajectory planningreplanning fora quadrotor unmanned aerial vehiclerdquo IEEE Transactions onAerospace and Electronic Systems vol 48 no 4 pp 2832ndash28482012

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[65] Y Chen and A L Schwartz ldquoRIOTS 95 aMATLAB toolbox forsolving general optimal control problems and its applicationsto chemical processesrdquo in Recent Developments in Optimizationand Optimal Control in Chemical Engineering R Luus Ed pp229ndash252 Transworld Research Publishers 2002

[66] I M Ross and M Karpenko ldquoA review of pseudospectraloptimal control from theory to flightrdquo Annual Reviews inControl vol 36 no 2 pp 182ndash197 2012

[67] J F Sturm ldquoUsing sedumi 102 a matlab toolbox for opti-mization over symmetric conesrdquo Optimization Methods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999

[68] Y V Pehlivanoglu ldquoA new vibrational genetic algorithmenhanced with a Voronoi diagram for path planning ofautonomous UAVrdquo Aerospace Science and Technology vol 16no 1 pp 47ndash55 2012

[69] G Erinc and S Carpin ldquoA genetic algorithm for nonholonomicmotion planningrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 1843ndash1849 RomaItaly April 2007

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[71] A Hacioglu and I Ozkol ldquoTransonic airfoil design andoptimisation by using vibrational genetic algorithmrdquo AircraftEngineering and Aerospace Technology vol 75 no 4 pp 350ndash357 2003

[72] A Hacioglu and I Ozkol ldquoVibrational genetic algorithm asa new concept in airfoil designrdquo Aircraft Engineering andAerospace Technology vol 74 no 3 pp 228ndash236 2002

[73] Y Chen X Zhao C Zhang and J Han ldquoRelative coordination3D trajectory generation based on the trimmed ACOrdquo inProceedings of the International Conference on Electrical and

Control Engineering (ICECE rsquo10) vol 1 pp 1531ndash1536 WuhanChina June 2010

[74] A Y Saber and A M Alshareef ldquoScalable unit commitmentby memory-bounded ant colony optimization with Alowast localsearchrdquo International Journal of Electrical Power and EnergySystems vol 30 no 6-7 pp 403ndash414 2008

[75] Y Del Valle G K Venayagamoorthy S Mohagheghi J-CHernandez and R G Harley ldquoParticle swarm optimizationbasic concepts variants and applications in power systemsrdquoIEEE Transactions on Evolutionary Computation vol 12 no 2pp 171ndash195 2008

[76] J Tang J Zhu and Z Sun ldquoA novel path planning approachbased on appart and particle swarm optimizationrdquo Advances inNeural Networks vol 3498 pp 253ndash258 2005

[77] H Pu Z Zhen and D Wang ldquoModified shuffled frog leapingalgorithm for optimization of UAV flight controllerrdquo Interna-tional Journal of Intelligent Computing and Cybernetics vol 4no 1 pp 25ndash39 2011

[78] A Rahimi-Vahed and A H Mirzaei ldquoA hybrid multi-objectiveshuffled frog-leaping algorithm for a mixed-model assemblyline sequencing problemrdquo Computers and Industrial Engineer-ing vol 53 no 4 pp 642ndash666 2007

[79] L Buriol P M Franca and P Moscato ldquoA new memeticalgorithm for the asymmetric traveling salesman problemrdquoJournal of Heuristics vol 10 no 5 pp 483ndash506 2004

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[81] V Kroumov and J Yu ldquo3D path planning for mobile robotsusing annealing neural networkrdquo in Proceedings of the IEEEInternational Conference on Networking Sensing and Control(ICNSC rsquo09) pp 130ndash135 Okayama Japan March 2009

[82] V Kroumov J Yu and K Shibayama ldquo3D path planningfor mobile robots using simulated annealing neural networkrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 7 pp 2885ndash2899 2010

[83] C W Ahn R S Ramakrishna C G Kang and I C ChoildquoShortest path routing algorithm using Hopfield neural net-workrdquo Electronics Letters vol 37 no 19 pp 1176ndash1178 2001

[84] S Jaishankar and R N Pralhad ldquo3D off-line path planningfor aerial vehicle using distance transform techniquerdquo ProcediaComputer Science vol 4 pp 1306ndash1315 2011

[85] F Schoslashler A la Cour-Harbo and M Bisgaard ldquoGeneratingapproximative minimum length paths in 3D for UAVsrdquo inProceedings of the IEEE Intelligent Vehicles Symposium (IV rsquo12)pp 229ndash233 Madrid Spain June 2012

[86] O Khatib Real-Time Obstacle Avoidance for Manipulators andMobile Robots Book Section 29 Springer New York NY USA1990

[87] R Deits and R Tedrake ldquoFootstep planning on uneven terrainwith mixed-integer convex optimizationrdquo in Proceedings of the14th IEEE-RAS International Conference on Humanoid Robots(Humanoids rsquo14) pp 279ndash286 Madrid Spain November 2014

[88] V A Luchnikov N N Medvedev L Oger and J-P TroadecldquoVoronoi-Delaunay analysis of voids in systems of nonsphericalparticlesrdquo Physical Review E vol 59 no 6 pp 7205ndash7212 1999

[89] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence University of Michigan Press Ann ArborMich USA 1975


[90] P Moscato and M G Norman ldquoA memetic approach for thetraveling salesman problem implementation of a computationalecology for combinatorial optimization on message-passingsystemsrdquo Parallel Computing and Transputer Applications vol1 pp 177ndash186 1992

[91] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning section 630 pp 760ndash766 Springer US 2010

[92] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[93] MM Eusuff and K E Lansey ldquoOptimization of water distribu-tion network design using the shuffled frog leaping algorithmrdquoJournal of Water Resources Planning and Management vol 129no 3 pp 210ndash225 2003

[94] S Chitta I Sucan and S Cousins ldquoMoveit[ros topics]rdquo IEEERobotics amp Automation Magazine vol 19 no 1 pp 18ndash19 2012

[95] L Kavraki and J-C Latombe ldquoRandomized preprocessing ofconfiguration space for fast path planningrdquo in Proceedings of theIEEE International Conference on Robotics and Automation vol3 pp 2138ndash2145 May 1994

[96] M I Shamos and D Hoey ldquoClosest-point problemsrdquo inProceedings of the 16th Annual Symposium on Foundations ofComputer Science pp 151ndash162 Berkeley Calif USA 1975

[97] F Sharifi Y Zhang and B W Gordon ldquoVoronoi-based cov-erage control for multi-quadrotor UAVsrdquo in Proceedings of theASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conference(IDETCCIE rsquo11) vol 6 pp 991ndash996 Washington DC USAAugust 2011

[98] R Geraerts ldquoPlanning short paths with clearance using explicitcorridorsrdquo in Proceedings of the IEEE International ConferenceonRobotics andAutomation (ICRA rsquo10) pp 1997ndash2004 Anchor-age Alaska USA 2010

[99] F B Zhan and C E Noon ldquoShortest path algorithms anevaluation using real road networksrdquo Transportation Sciencevol 32 no 1 pp 65ndash73 1998

[100] L Verscheure L Peyrodie N Makni N Betrouni S Maoucheand M Vermandel ldquoDijkstrarsquos algorithm applied to 3D skele-tonization of the brain vascular tree evaluation and applica-tion to symbolicrdquo in Proceedings of the Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety (EMBC rsquo10) pp 3081ndash3084 Buenos Aires ArgentinaSeptember 2010

[101] M Wan Z Liang Q Ke L Hong I Bitter and A KaufmanldquoAutomatic centerline extraction for virtual colonoscopyrdquo IEEETransactions on Medical Imaging vol 21 no 12 pp 1450ndash14602002

[102] I A Musliman A A Rahman and V Coors ldquoImplementing3d network analysis in 3d-gisrdquo in Proceedings of the 21st ISPRSCongress Silk Road for Information from Imagery vol 8 pp 113ndash120 Beijing China 2008

[103] L De Filippis G Guglieri and F Quagliotti ldquoPath planningstrategies for UAVS in 3D environmentsrdquo Journal of Intelligentand Robotic Systems vol 65 no 1ndash4 pp 247ndash264 2012

[104] C Tovey S Greenberg and S Koenig ldquoImproved analysisof Dlowastrdquo in Proceedings of the IEEE International Conferenceon Robotics and Automation pp 3371ndash3378 Taipei TaiwanSeptember 2003

[105] C L Shih T-T Lee and W A Gruver ldquoA unified approach forrobotmotion planningwithmoving polyhedral obstaclesrdquo IEEE

Transactions on Systems Man and Cybernetics vol 20 no 4 pp903ndash915 1990

[106] K Culligan M Valenti Y Kuwata and J P How ldquoThree-dimensional flight experiments using on-line mixed-integerlinear programming trajectory optimizationrdquo in Proceedings ofthe American Control Conference (ACC rsquo07) vol 6 pp 5322ndash5327 New York NY USA July 2007

[107] S J Anderson S C Peters T E Pilutti and K IagnemmaldquoAn optimal-control-based framework for trajectory planningthreat assessment and semi-autonomous control of passengervehicles in hazard avoidance Scenariosrdquo International Journal ofVehicle Autonomous Systems vol 8 no 2ndash4 pp 190ndash216 2010

[108] C Tricaud and Y-Q Chen ldquoLinear and nonlinear modelpredictive control using a general purpose optimal controlproblem solver riots 95rdquo in Proceedings of the Chinese Controland Decision Conference pp 1552ndash1557 Yantai China 2008

[109] J Dong and V Juris Parallel Evolutionary Algorithms for UAVPath Planning section 1 American Institute of Aeronautics andAstronautics 2004

[110] M Dorigo and M Birattari ldquoAnt colony optimizationrdquo inEncyclopedia ofMachine Learning pp 36ndash39 SpringerUS 2010

[111] S Kundu and D R Parhi ldquoModified shuffled frog leapingalgorithm based 6DOF motion for underwater mobile robotrdquoProcedia Technology vol 10 pp 295ndash303 2013

[112] F Jung Leng K Jared O James and W Eliot Three-Dimensional Path Planning of Unmanned Aerial Vehicles UsingParticle SwarmOptimization American Institute ofAeronauticsand Astronautics 2006

[113] R Glasius A Komoda and S C AM Gielen ldquoNeural networkdynamics for path planning and obstacle avoidancerdquo NeuralNetworks vol 8 no 1 pp 125ndash133 1995

[114] A A Kassim and B V K V Kumar ldquoA neural network architec-ture for path planningrdquo in Proceedings of the International JointConference on Neural Networks vol 2 pp 787ndash792 BaltimoreMd USA 1992

[115] C Luo and S X Yang ldquoA real-time cooperative sweepingstrategy formultiple cleaning robotsrdquo in Proceedings of the IEEEInternational Symposium on Intelligent Control pp 660ndash665Vancouver Canada October 2002

[116] S X Yang and C Luo ldquoA neural network approach to completecoverage path planningrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 34 no 1 pp 718ndash7252004

[117] X Zhang H Duan and Y Yu ldquoReceding horizon controlfor multi-UAVs close formation control based on differentialevolutionrdquo Science China Information Sciences vol 53 no 2 pp223ndash235 2010

[118] D Theilliol httppage-persocranuhp-nancyfrsitejoomla

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



(a) Forest [15] (b) Urban [16] (c) Underwater [17]

Figure 1 Examples of 3D complex environments

to sampling based algorithm to explain the 3D planningidea

Facing the more and more challenging environment thatrobots faced where environments tend to be unstructuredand full of uncertain factors 3D path planning algorithmsare urgently needed nowadays Several typical kinds of 3Denvironments are presented in Figure 1 including foresturban and underwater environments When planning inthese complex situations a simple 2D algorithm will not bequalified thus 3D path planning algorithms are needed

Path planning in 3D environment shows great prospectbut unlike 2D path planning the difficulties increase expo-nentially with kinematic constraints In order to plan acollision-free path through the cluster environment theproblem of how to model the environment while takingthe kinematic constraints into consideration needs to besolved From the optimization point of view finding a 3Doptimal path planning problem is NP-hard thus there existno common solutions

3D path planning algorithms include visibility graph [18]which works by connecting visible vertexes of polyhedronrandom-exploring algorithms such as rapidly exploring ran-dom tree [19] Probabilistic Road Map [20] optimal searchalgorithms (such as Dijkstrarsquos algorithm [21] Alowast [22] andDlowast [23]) and bioinspired planning algorithmsWhatmust beemphasized is that this paper only pays attention to broadlyapplicable methods proposed Algorithms such as manifoldbased algorithms [24] which generate smooth path havebeen ignored due to reason that it is only applicable torigid body robots without aerodynamical or hydromechan-ical influences Synthesizing all these methods a two-stepprocedure of 3D path planning is summarized as follows

Step 1 The first step is environment perception and model-ing usually using a grid map [25] (with occupancy proba-bility) or a convex or nonconvex region in combination ofpolynomial form [26]

Step 2 Then path planning algorithm is employed to find thebest path according to the cost function with the ability toachieve both time efficiency and cost minimum

Steps above are not absolute procedures we must obeyand Step 1 can be divided into two parts or Step 2 integrateswith Step 1 According to Step 2 this paper discusses theefficiency of each algorithm and particularly concentrateson time complexity and applicability Path planning in 3Denvironments may face much more uncertainties thus allthis should be taken into consideration to achieve the best

path Local minimal and global optimal are two contradictivepoints this paper will analyze these two properties of thealgorithms This paper mainly answered several importantquestions (a) what is the taxonomy of these 3D path planningalgorithms (b) How do these algorithms work in order tofind the path (c) Why is it suitable or not suitable for suchenvironment

This paper is an extension of [27] with more comprehen-sive explanation on each kind of algorithms

The main contributions of this paper are listed below(1)Thefirst contribution is proposing a taxonomymethod

for 3D path planning algorithms and puts forward a newcategory called multifusion based algorithms

(2) The second contribution of this paper is providing acomprehensive analysis of 3D path planning algorithmwhichcontains almost all of the current most successful methods

The following sections are arranged as follows Sec-tion 2 discusses some controversial points which need tobe declared and gives definitions to these problems forfurther discussion Section 3 explains the taxonomy of 3Dpath planning algorithms and gives a detailed analysis ofthe taxonomyrsquos reason and also lists elements of each cate-gory This section includes the basic concept of each kindof algorithms A series of sampling based algorithms arediscussed particularly and each element is put forward bycomparison in Section 4 Node based optimal algorithmsrsquocommon properties are analyzed in Section 5 and thiskind of algorithms shares the same merits In Section 6this paper discusses a special kind of planning algorithmsthat is mathematic model based algorithms and confirmsit to be path planning algorithm Section 7 concentrateson the working mechanism of bioinspired algorithms andthree typical algorithms are analyzed elaborately Multifusionbased algorithms are discussed in Section 8 which arecommonly ignored Some typical researches are listed toprove this categoryThe last section outlines a conclusion andsome directions for further research

2 Preliminary Materials

This section discusses some controversial or ignored pointswhich needed to be declared for further discussion inthe following sections namely the difference between pathplanning and optimal path planning and definition of pathplanning and trajectory planning

21 Problem Statement Robots have the ability to operatewithout (or just a little guiding from) human Ranging from



Samplingbased

algorithms

basedNode

optimalalgorithms

modelMathmtic

basedalgorithms


Multifusionbased

algorithms











1015840 and 119862(1205751015840

) =













Active Passive

RRTlowast

RRTDDRRT

and so forth

3DVoronoi

RRGand so forth

RMlowast) and

PRMk-PRMs-PRM(P

so forth

Artificial

Fieldand so forth

Potential




and so forth

Dlowast


Alowast LPAlowast

lowastlazy Thetaand










Linear algorithms

NLP

Optimal control

NLP



and soMILP

forth

BIPand soforth


Neural network

GAand soforth

SFLAand soforth


and soforth

MAand soforth

PSOand soforth

ACO









41 RRT Series






xinit

xrandom1

xrandom2

xnew1

xnew2xnear

120574

















xinit


xinit




xinit


xinit






x1

x2

x3

xnearest

xinit


x1

x2

x3

xinit



















119899


1015840


1015840

= 1198751015840

cup 119909new(6) 119864

1015840

= 1198641015840


1015840

|)


1015840

= 1198641015840




1015840


1015840

= 1198751015840

cup 119909near(6) 119864

1015840

= 1198641015840




1015840

= 1198641015840


(19) 1198641015840

= 1198641015840


(20) End(21) EndReturn 1198661015840 = (1198751015840 1198641015840)





119889119894(119903) = 119889

119895(119903) = 119889

119896(119903) (1)





1015840




119889119894(119903 + V) = 119889

119895(119903 + V) = 119889

119896(119903 + V) (2)



119895sdot V)

10038161003816100381610038161003816119903= (nabla119889

119896sdot V)1003816100381610038161003816119903 (3)



Φ = (119889119894minus 119889119895)2

= (119889119894minus 119889119896)2

= (119889119895minus 119889119896)2

(4)







119880119905(119883) = 119880

119886(119883) + 119880

119903(119883) (5)



119903




119891119905(119883) = minusnabla119880

119905= 119891119886(119883) +

allsum

119894=1

119891119903119894(119883) (6)



119880119886= 119896119886

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817 (7)

119891119886(119883) = minusnabla119880

119886= minus119896119886

(119883119866minus 119883)

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817

(8)

where 119883119866




119880119903119894=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894)) if 120578

119894(119883) le 120578

0119894

(9)

119891119903119894(119883) = minusnabla119880

119903119894(119883)

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894

1205782

119894(119883)

(1

120578119894(119883)

minus1

1205780(119894))nabla120578119894(119883) if 120578

119894(119883) le 120578

0119894

(10)



119894(119883) is a polar value 119891

119903119894is the 119894th




min119906

⟨119881 (119909) 119906⟩ = minus1 (11)


119880119903119894

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894))

2

120588119899

(119883119883119866) if 120578

119894(119883) le 120578

0119894

(12)


and the goal119883119866 120588119899(119883119883


119886(119883)
























1199091015840


119868)


1015840

larr 119891(119909 119906)

(7) if 119862(119909) + 119897(119909 119906) lt min119862(1199091015840) 119862(119909119866)

(8) 1198621015840

(119909) = 119862(119909) + 119897(119909 119906)

(9) if 1199091015840 = 119909119866





119891 (119899) = 119892 (119909) + ℎ (119909) (13)













1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)


1015840










119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)



Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Samplingbased

algorithms

basedNode

optimalalgorithms

modelMathmtic

basedalgorithms


Multifusionbased

algorithms











1015840 and 119862(1205751015840

) =













Active Passive

RRTlowast

RRTDDRRT

and so forth

3DVoronoi

RRGand so forth

RMlowast) and

PRMk-PRMs-PRM(P

so forth

Artificial

Fieldand so forth

Potential




and so forth

Dlowast


Alowast LPAlowast

lowastlazy Thetaand










Linear algorithms

NLP

Optimal control

NLP



and soMILP

forth

BIPand soforth


Neural network

GAand soforth

SFLAand soforth


and soforth

MAand soforth

PSOand soforth

ACO









41 RRT Series






xinit

xrandom1

xrandom2

xnew1

xnew2xnear

120574

















xinit


xinit




xinit


xinit






x1

x2

x3

xnearest

xinit


x1

x2

x3

xinit



















119899


1015840


1015840

= 1198751015840

cup 119909new(6) 119864

1015840

= 1198641015840


1015840

|)


1015840

= 1198641015840




1015840


1015840

= 1198751015840

cup 119909near(6) 119864

1015840

= 1198641015840




1015840

= 1198641015840


(19) 1198641015840

= 1198641015840


(20) End(21) EndReturn 1198661015840 = (1198751015840 1198641015840)





119889119894(119903) = 119889

119895(119903) = 119889

119896(119903) (1)





1015840




119889119894(119903 + V) = 119889

119895(119903 + V) = 119889

119896(119903 + V) (2)



119895sdot V)

10038161003816100381610038161003816119903= (nabla119889

119896sdot V)1003816100381610038161003816119903 (3)



Φ = (119889119894minus 119889119895)2

= (119889119894minus 119889119896)2

= (119889119895minus 119889119896)2

(4)







119880119905(119883) = 119880

119886(119883) + 119880

119903(119883) (5)



119903




119891119905(119883) = minusnabla119880

119905= 119891119886(119883) +

allsum

119894=1

119891119903119894(119883) (6)



119880119886= 119896119886

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817 (7)

119891119886(119883) = minusnabla119880

119886= minus119896119886

(119883119866minus 119883)

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817

(8)

where 119883119866




119880119903119894=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894)) if 120578

119894(119883) le 120578

0119894

(9)

119891119903119894(119883) = minusnabla119880

119903119894(119883)

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894

1205782

119894(119883)

(1

120578119894(119883)

minus1

1205780(119894))nabla120578119894(119883) if 120578

119894(119883) le 120578

0119894

(10)



119894(119883) is a polar value 119891

119903119894is the 119894th




min119906

⟨119881 (119909) 119906⟩ = minus1 (11)


119880119903119894

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894))

2

120588119899

(119883119883119866) if 120578

119894(119883) le 120578

0119894

(12)


and the goal119883119866 120588119899(119883119883


119886(119883)
























1199091015840


119868)


1015840

larr 119891(119909 119906)

(7) if 119862(119909) + 119897(119909 119906) lt min119862(1199091015840) 119862(119909119866)

(8) 1198621015840

(119909) = 119862(119909) + 119897(119909 119906)

(9) if 1199091015840 = 119909119866





119891 (119899) = 119892 (119909) + ℎ (119909) (13)













1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)


1015840










119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)



Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Active Passive

RRTlowast

RRTDDRRT

and so forth

3DVoronoi

RRGand so forth

RMlowast) and

PRMk-PRMs-PRM(P

so forth

Artificial

Fieldand so forth

Potential




and so forth

Dlowast


Alowast LPAlowast

lowastlazy Thetaand










Linear algorithms

NLP

Optimal control

NLP



and soMILP

forth

BIPand soforth


Neural network

GAand soforth

SFLAand soforth


and soforth

MAand soforth

PSOand soforth

ACO









41 RRT Series






xinit

xrandom1

xrandom2

xnew1

xnew2xnear

120574

















xinit


xinit




xinit


xinit






x1

x2

x3

xnearest

xinit


x1

x2

x3

xinit



















119899


1015840


1015840

= 1198751015840

cup 119909new(6) 119864

1015840

= 1198641015840


1015840

|)


1015840

= 1198641015840




1015840


1015840

= 1198751015840

cup 119909near(6) 119864

1015840

= 1198641015840




1015840

= 1198641015840


(19) 1198641015840

= 1198641015840


(20) End(21) EndReturn 1198661015840 = (1198751015840 1198641015840)





119889119894(119903) = 119889

119895(119903) = 119889

119896(119903) (1)





1015840




119889119894(119903 + V) = 119889

119895(119903 + V) = 119889

119896(119903 + V) (2)



119895sdot V)

10038161003816100381610038161003816119903= (nabla119889

119896sdot V)1003816100381610038161003816119903 (3)



Φ = (119889119894minus 119889119895)2

= (119889119894minus 119889119896)2

= (119889119895minus 119889119896)2

(4)







119880119905(119883) = 119880

119886(119883) + 119880

119903(119883) (5)



119903




119891119905(119883) = minusnabla119880

119905= 119891119886(119883) +

allsum

119894=1

119891119903119894(119883) (6)



119880119886= 119896119886

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817 (7)

119891119886(119883) = minusnabla119880

119886= minus119896119886

(119883119866minus 119883)

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817

(8)

where 119883119866




119880119903119894=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894)) if 120578

119894(119883) le 120578

0119894

(9)

119891119903119894(119883) = minusnabla119880

119903119894(119883)

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894

1205782

119894(119883)

(1

120578119894(119883)

minus1

1205780(119894))nabla120578119894(119883) if 120578

119894(119883) le 120578

0119894

(10)



119894(119883) is a polar value 119891

119903119894is the 119894th




min119906

⟨119881 (119909) 119906⟩ = minus1 (11)


119880119903119894

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894))

2

120588119899

(119883119883119866) if 120578

119894(119883) le 120578

0119894

(12)


and the goal119883119866 120588119899(119883119883


119886(119883)
























1199091015840


119868)


1015840

larr 119891(119909 119906)

(7) if 119862(119909) + 119897(119909 119906) lt min119862(1199091015840) 119862(119909119866)

(8) 1198621015840

(119909) = 119862(119909) + 119897(119909 119906)

(9) if 1199091015840 = 119909119866





119891 (119899) = 119892 (119909) + ℎ (119909) (13)













1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)


1015840










119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)



Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Linear algorithms

NLP

Optimal control

NLP



and soMILP

forth

BIPand soforth


Neural network

GAand soforth

SFLAand soforth


and soforth

MAand soforth

PSOand soforth

ACO









41 RRT Series






xinit

xrandom1

xrandom2

xnew1

xnew2xnear

120574

















xinit


xinit




xinit


xinit






x1

x2

x3

xnearest

xinit


x1

x2

x3

xinit



















119899


1015840


1015840

= 1198751015840

cup 119909new(6) 119864

1015840

= 1198641015840


1015840

|)


1015840

= 1198641015840




1015840


1015840

= 1198751015840

cup 119909near(6) 119864

1015840

= 1198641015840




1015840

= 1198641015840


(19) 1198641015840

= 1198641015840


(20) End(21) EndReturn 1198661015840 = (1198751015840 1198641015840)





119889119894(119903) = 119889

119895(119903) = 119889

119896(119903) (1)





1015840




119889119894(119903 + V) = 119889

119895(119903 + V) = 119889

119896(119903 + V) (2)



119895sdot V)

10038161003816100381610038161003816119903= (nabla119889

119896sdot V)1003816100381610038161003816119903 (3)



Φ = (119889119894minus 119889119895)2

= (119889119894minus 119889119896)2

= (119889119895minus 119889119896)2

(4)







119880119905(119883) = 119880

119886(119883) + 119880

119903(119883) (5)



119903




119891119905(119883) = minusnabla119880

119905= 119891119886(119883) +

allsum

119894=1

119891119903119894(119883) (6)



119880119886= 119896119886

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817 (7)

119891119886(119883) = minusnabla119880

119886= minus119896119886

(119883119866minus 119883)

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817

(8)

where 119883119866




119880119903119894=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894)) if 120578

119894(119883) le 120578

0119894

(9)

119891119903119894(119883) = minusnabla119880

119903119894(119883)

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894

1205782

119894(119883)

(1

120578119894(119883)

minus1

1205780(119894))nabla120578119894(119883) if 120578

119894(119883) le 120578

0119894

(10)



119894(119883) is a polar value 119891

119903119894is the 119894th




min119906

⟨119881 (119909) 119906⟩ = minus1 (11)


119880119903119894

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894))

2

120588119899

(119883119883119866) if 120578

119894(119883) le 120578

0119894

(12)


and the goal119883119866 120588119899(119883119883


119886(119883)
























1199091015840


119868)


1015840

larr 119891(119909 119906)

(7) if 119862(119909) + 119897(119909 119906) lt min119862(1199091015840) 119862(119909119866)

(8) 1198621015840

(119909) = 119862(119909) + 119897(119909 119906)

(9) if 1199091015840 = 119909119866





119891 (119899) = 119892 (119909) + ℎ (119909) (13)













1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)


1015840










119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)



Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










xinit

xrandom1

xrandom2

xnew1

xnew2xnear

120574

















xinit


xinit




xinit


xinit






x1

x2

x3

xnearest

xinit


x1

x2

x3

xinit



















119899


1015840


1015840

= 1198751015840

cup 119909new(6) 119864

1015840

= 1198641015840


1015840

|)


1015840

= 1198641015840




1015840


1015840

= 1198751015840

cup 119909near(6) 119864

1015840

= 1198641015840




1015840

= 1198641015840


(19) 1198641015840

= 1198641015840


(20) End(21) EndReturn 1198661015840 = (1198751015840 1198641015840)





119889119894(119903) = 119889

119895(119903) = 119889

119896(119903) (1)





1015840




119889119894(119903 + V) = 119889

119895(119903 + V) = 119889

119896(119903 + V) (2)



119895sdot V)

10038161003816100381610038161003816119903= (nabla119889

119896sdot V)1003816100381610038161003816119903 (3)



Φ = (119889119894minus 119889119895)2

= (119889119894minus 119889119896)2

= (119889119895minus 119889119896)2

(4)







119880119905(119883) = 119880

119886(119883) + 119880

119903(119883) (5)



119903




119891119905(119883) = minusnabla119880

119905= 119891119886(119883) +

allsum

119894=1

119891119903119894(119883) (6)



119880119886= 119896119886

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817 (7)

119891119886(119883) = minusnabla119880

119886= minus119896119886

(119883119866minus 119883)

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817

(8)

where 119883119866




119880119903119894=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894)) if 120578

119894(119883) le 120578

0119894

(9)

119891119903119894(119883) = minusnabla119880

119903119894(119883)

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894

1205782

119894(119883)

(1

120578119894(119883)

minus1

1205780(119894))nabla120578119894(119883) if 120578

119894(119883) le 120578

0119894

(10)



119894(119883) is a polar value 119891

119903119894is the 119894th




min119906

⟨119881 (119909) 119906⟩ = minus1 (11)


119880119903119894

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894))

2

120588119899

(119883119883119866) if 120578

119894(119883) le 120578

0119894

(12)


and the goal119883119866 120588119899(119883119883


119886(119883)
























1199091015840


119868)


1015840

larr 119891(119909 119906)

(7) if 119862(119909) + 119897(119909 119906) lt min119862(1199091015840) 119862(119909119866)

(8) 1198621015840

(119909) = 119862(119909) + 119897(119909 119906)

(9) if 1199091015840 = 119909119866





119891 (119899) = 119892 (119909) + ℎ (119909) (13)













1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)


1015840










119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)



Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










xinit


xinit




xinit


xinit






x1

x2

x3

xnearest

xinit


x1

x2

x3

xinit



















119899


1015840


1015840

= 1198751015840

cup 119909new(6) 119864

1015840

= 1198641015840


1015840

|)


1015840

= 1198641015840




1015840


1015840

= 1198751015840

cup 119909near(6) 119864

1015840

= 1198641015840




1015840

= 1198641015840


(19) 1198641015840

= 1198641015840


(20) End(21) EndReturn 1198661015840 = (1198751015840 1198641015840)





119889119894(119903) = 119889

119895(119903) = 119889

119896(119903) (1)





1015840




119889119894(119903 + V) = 119889

119895(119903 + V) = 119889

119896(119903 + V) (2)



119895sdot V)

10038161003816100381610038161003816119903= (nabla119889

119896sdot V)1003816100381610038161003816119903 (3)



Φ = (119889119894minus 119889119895)2

= (119889119894minus 119889119896)2

= (119889119895minus 119889119896)2

(4)







119880119905(119883) = 119880

119886(119883) + 119880

119903(119883) (5)



119903




119891119905(119883) = minusnabla119880

119905= 119891119886(119883) +

allsum

119894=1

119891119903119894(119883) (6)



119880119886= 119896119886

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817 (7)

119891119886(119883) = minusnabla119880

119886= minus119896119886

(119883119866minus 119883)

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817

(8)

where 119883119866




119880119903119894=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894)) if 120578

119894(119883) le 120578

0119894

(9)

119891119903119894(119883) = minusnabla119880

119903119894(119883)

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894

1205782

119894(119883)

(1

120578119894(119883)

minus1

1205780(119894))nabla120578119894(119883) if 120578

119894(119883) le 120578

0119894

(10)



119894(119883) is a polar value 119891

119903119894is the 119894th




min119906

⟨119881 (119909) 119906⟩ = minus1 (11)


119880119903119894

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894))

2

120588119899

(119883119883119866) if 120578

119894(119883) le 120578

0119894

(12)


and the goal119883119866 120588119899(119883119883


119886(119883)
























1199091015840


119868)


1015840

larr 119891(119909 119906)

(7) if 119862(119909) + 119897(119909 119906) lt min119862(1199091015840) 119862(119909119866)

(8) 1198621015840

(119909) = 119862(119909) + 119897(119909 119906)

(9) if 1199091015840 = 119909119866





119891 (119899) = 119892 (119909) + ℎ (119909) (13)













1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)


1015840










119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)



Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x1

x2

x3

xnearest

xinit


x1

x2

x3

xinit



















119899


1015840


1015840

= 1198751015840

cup 119909new(6) 119864

1015840

= 1198641015840


1015840

|)


1015840

= 1198641015840




1015840


1015840

= 1198751015840

cup 119909near(6) 119864

1015840

= 1198641015840




1015840

= 1198641015840


(19) 1198641015840

= 1198641015840


(20) End(21) EndReturn 1198661015840 = (1198751015840 1198641015840)





119889119894(119903) = 119889

119895(119903) = 119889

119896(119903) (1)





1015840




119889119894(119903 + V) = 119889

119895(119903 + V) = 119889

119896(119903 + V) (2)



119895sdot V)

10038161003816100381610038161003816119903= (nabla119889

119896sdot V)1003816100381610038161003816119903 (3)



Φ = (119889119894minus 119889119895)2

= (119889119894minus 119889119896)2

= (119889119895minus 119889119896)2

(4)







119880119905(119883) = 119880

119886(119883) + 119880

119903(119883) (5)



119903




119891119905(119883) = minusnabla119880

119905= 119891119886(119883) +

allsum

119894=1

119891119903119894(119883) (6)



119880119886= 119896119886

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817 (7)

119891119886(119883) = minusnabla119880

119886= minus119896119886

(119883119866minus 119883)

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817

(8)

where 119883119866




119880119903119894=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894)) if 120578

119894(119883) le 120578

0119894

(9)

119891119903119894(119883) = minusnabla119880

119903119894(119883)

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894

1205782

119894(119883)

(1

120578119894(119883)

minus1

1205780(119894))nabla120578119894(119883) if 120578

119894(119883) le 120578

0119894

(10)



119894(119883) is a polar value 119891

119903119894is the 119894th




min119906

⟨119881 (119909) 119906⟩ = minus1 (11)


119880119903119894

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894))

2

120588119899

(119883119883119866) if 120578

119894(119883) le 120578

0119894

(12)


and the goal119883119866 120588119899(119883119883


119886(119883)
























1199091015840


119868)


1015840

larr 119891(119909 119906)

(7) if 119862(119909) + 119897(119909 119906) lt min119862(1199091015840) 119862(119909119866)

(8) 1198621015840

(119909) = 119862(119909) + 119897(119909 119906)

(9) if 1199091015840 = 119909119866





119891 (119899) = 119892 (119909) + ℎ (119909) (13)













1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)


1015840










119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)



Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119899


1015840


1015840

= 1198751015840

cup 119909new(6) 119864

1015840

= 1198641015840


1015840

|)


1015840

= 1198641015840




1015840


1015840

= 1198751015840

cup 119909near(6) 119864

1015840

= 1198641015840




1015840

= 1198641015840


(19) 1198641015840

= 1198641015840


(20) End(21) EndReturn 1198661015840 = (1198751015840 1198641015840)





119889119894(119903) = 119889

119895(119903) = 119889

119896(119903) (1)





1015840




119889119894(119903 + V) = 119889

119895(119903 + V) = 119889

119896(119903 + V) (2)



119895sdot V)

10038161003816100381610038161003816119903= (nabla119889

119896sdot V)1003816100381610038161003816119903 (3)



Φ = (119889119894minus 119889119895)2

= (119889119894minus 119889119896)2

= (119889119895minus 119889119896)2

(4)







119880119905(119883) = 119880

119886(119883) + 119880

119903(119883) (5)



119903




119891119905(119883) = minusnabla119880

119905= 119891119886(119883) +

allsum

119894=1

119891119903119894(119883) (6)



119880119886= 119896119886

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817 (7)

119891119886(119883) = minusnabla119880

119886= minus119896119886

(119883119866minus 119883)

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817

(8)

where 119883119866




119880119903119894=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894)) if 120578

119894(119883) le 120578

0119894

(9)

119891119903119894(119883) = minusnabla119880

119903119894(119883)

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894

1205782

119894(119883)

(1

120578119894(119883)

minus1

1205780(119894))nabla120578119894(119883) if 120578

119894(119883) le 120578

0119894

(10)



119894(119883) is a polar value 119891

119903119894is the 119894th




min119906

⟨119881 (119909) 119906⟩ = minus1 (11)


119880119903119894

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894))

2

120588119899

(119883119883119866) if 120578

119894(119883) le 120578

0119894

(12)


and the goal119883119866 120588119899(119883119883


119886(119883)
























1199091015840


119868)


1015840

larr 119891(119909 119906)

(7) if 119862(119909) + 119897(119909 119906) lt min119862(1199091015840) 119862(119909119866)

(8) 1198621015840

(119909) = 119862(119909) + 119897(119909 119906)

(9) if 1199091015840 = 119909119866





119891 (119899) = 119892 (119909) + ℎ (119909) (13)













1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)


1015840










119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)



Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1015840




119889119894(119903 + V) = 119889

119895(119903 + V) = 119889

119896(119903 + V) (2)



119895sdot V)

10038161003816100381610038161003816119903= (nabla119889

119896sdot V)1003816100381610038161003816119903 (3)



Φ = (119889119894minus 119889119895)2

= (119889119894minus 119889119896)2

= (119889119895minus 119889119896)2

(4)







119880119905(119883) = 119880

119886(119883) + 119880

119903(119883) (5)



119903




119891119905(119883) = minusnabla119880

119905= 119891119886(119883) +

allsum

119894=1

119891119903119894(119883) (6)



119880119886= 119896119886

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817 (7)

119891119886(119883) = minusnabla119880

119886= minus119896119886

(119883119866minus 119883)

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817

(8)

where 119883119866




119880119903119894=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894)) if 120578

119894(119883) le 120578

0119894

(9)

119891119903119894(119883) = minusnabla119880

119903119894(119883)

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894

1205782

119894(119883)

(1

120578119894(119883)

minus1

1205780(119894))nabla120578119894(119883) if 120578

119894(119883) le 120578

0119894

(10)



119894(119883) is a polar value 119891

119903119894is the 119894th




min119906

⟨119881 (119909) 119906⟩ = minus1 (11)


119880119903119894

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894))

2

120588119899

(119883119883119866) if 120578

119894(119883) le 120578

0119894

(12)


and the goal119883119866 120588119899(119883119883


119886(119883)
























1199091015840


119868)


1015840

larr 119891(119909 119906)

(7) if 119862(119909) + 119897(119909 119906) lt min119862(1199091015840) 119862(119909119866)

(8) 1198621015840

(119909) = 119862(119909) + 119897(119909 119906)

(9) if 1199091015840 = 119909119866





119891 (119899) = 119892 (119909) + ℎ (119909) (13)













1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)


1015840










119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)



Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119880119886= 119896119886

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817 (7)

119891119886(119883) = minusnabla119880

119886= minus119896119886

(119883119866minus 119883)

1003817100381710038171003817119883119866 minus 1198831003817100381710038171003817

(8)

where 119883119866




119880119903119894=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894)) if 120578

119894(119883) le 120578

0119894

(9)

119891119903119894(119883) = minusnabla119880

119903119894(119883)

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894

1205782

119894(119883)

(1

120578119894(119883)

minus1

1205780(119894))nabla120578119894(119883) if 120578

119894(119883) le 120578

0119894

(10)



119894(119883) is a polar value 119891

119903119894is the 119894th




min119906

⟨119881 (119909) 119906⟩ = minus1 (11)


119880119903119894

=

0 if 120578119894(119883) gt 120578

0119894

119870119903119894(

1

120578119894(119883)

minus1

1205780(119894))

2

120588119899

(119883119883119866) if 120578

119894(119883) le 120578

0119894

(12)


and the goal119883119866 120588119899(119883119883


119886(119883)
























1199091015840


119868)


1015840

larr 119891(119909 119906)

(7) if 119862(119909) + 119897(119909 119906) lt min119862(1199091015840) 119862(119909119866)

(8) 1198621015840

(119909) = 119862(119909) + 119897(119909 119906)

(9) if 1199091015840 = 119909119866





119891 (119899) = 119892 (119909) + ℎ (119909) (13)













1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)


1015840










119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)



Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























1199091015840


119868)


1015840

larr 119891(119909 119906)

(7) if 119862(119909) + 119897(119909 119906) lt min119862(1199091015840) 119862(119909119866)

(8) 1198621015840

(119909) = 119862(119909) + 119897(119909 119906)

(9) if 1199091015840 = 119909119866





119891 (119899) = 119892 (119909) + ℎ (119909) (13)













1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)


1015840










119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)



Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















1198911015840

(119899) = 1198921015840

(119909) + ℎ1015840

(119909) (14)


1015840










119869cost-goal = 119891 (119906 119909 119910 119911) + 120593 (119909 119910 119911) + 119877 (119909 119910 119911) (15)

subject to

1205950= 120595 [119906 (119905

0) 119909 (119905

0) 119910 (119905

0) 119911 (119905

0)] (16)

120595119891= 120595 [119906 (119905

119891) 119909 (119905

119891) 119910 (119905

119891) 119911 (119905

119891)] (17)

119892lower le 119892 (119906 119909 119910 119911) le 119892upper (18)

119906lower le 119906 (119905) le 119906upper (19)



Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Maximumprinciple



Min goalconditions








= 119891 (119909 (119905) 119906 (119905)) (20)


119871 = 120601 [119909 (119905) 119906 (119905)] (21)


120595 (119909 (1199050) 1199050) = 119886 (22)

120595 (119909 (119905119891) 119905119891) = 119887 (23)


119869 = 120601 [119909 (119905) 119906 (119905)] + int

119905119891

1199050

120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] minus (24)


119867 = 120582119879

(119905) 119891 [119909 (119905) 119906 (119905)] (25)



















119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























119879119894119895(119905 + 119899) = 120588 sdot 119879

119894119895(119905) + Δ119879

119894119895 (26)



Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Initialization

Selectparents

Decode Path

Mutationcrossover




Δ119879119894119895=

119898

sum

119896=1

Δ119879119896

119894119895

Δ119879119896

119894119895=

119876

119871119896


0 if otherwise

(27)

where Δ119879119894119895


119894119895is the 119896th ant



120578119894119895=

1

119889119894119895

(28)

where 119889119894119895








119889119909119894

119889119905= minus119860119909

119894+ (119861 minus 119909

119894)([119868119894]+

+

119896

sum

119895=1

119908119894119895[119909119895]+

)

minus (119863 + 119909119894) [119868119894]minus

(29)



119894]+

+ sum119896

119895=1119908119894119895[119909119895]+ and













































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















































2


2


2



Competing Interests


Acknowledgments


References






























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









review article survey of robot 3d path planning...

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