maximizing the coverage of roadmap graph for optimal...

Research ArticleMaximizing the Coverage of Roadmap Graph forOptimal Motion Planning

Jae-Han Park 1 and Tae-Woong Yoon 2

1Robotics RampD Group Korea Institute of Industrial Technology (KITECH) Ansan Republic of Korea2School of Electrical Engineering Korea University Seoul Republic of Korea

Correspondence should be addressed to Tae-Woong Yoon twykoreaackr

Received 26 April 2018 Revised 15 September 2018 Accepted 4 October 2018 Published 8 November 2018

Academic Editor Zhiwei Gao

Copyright copy 2018 Jae-Han Park and Tae-Woong Yoon This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Automated motion-planning technologies for industrial robots are critical for their application to Industry 40 Various sampling-based methods have been studied to generate the collision-free motion of articulated industrial robots Such sampling-basedmethods provide efficient solutions to complex planning problems but their limitations hinder the attainment of optimal resultsThis paper considers a method to obtain the optimal results in the roadmap algorithm that is representative of the sampling-basedmethod We define the coverage of a graph as a performance index of its optimality as constructed by a sampling-based algorithmand propose an optimization algorithm that can maximize graph coverage in the configuration space The proposed method wasapplied to the model of an industrial robot and the results of the simulation confirm that the roadmap graph obtained by theproposed algorithm can generate results of satisfactory quality in path-finding tests under various conditions

1 Introduction

Automatic motion planning for robots is an importanttechnology for next-generation manufacturing systems suchas Industry 40 Sampling-based algorithms have been widelyused in robot motion-planning problems because they areefficient at providing reasonable solutions [1ndash4] Sampling-based algorithms determine feasible paths for the robotrsquosmotion using information from a graph that consists ofrandomly sampled nodes and connected edges in the givenconfiguration space Such randomized approaches have astrong advantage in terms of quickly providing solutions tocomplex problems such as in a high-dimensional configu-ration space However sampling-based methods guaranteeprobabilistic completeness and thus they require a largenumber of sampling nodes to find paths for the motion of therobot in arbitrary cases [2]The probabilistic roadmap (PRM)[3] and rapidly exploring random trees (RRT) [5] are represen-tative of such sampling-based motion-planning algorithms

The PRM method is composed of learning and queryphases In the query phase collision-free paths for themotion

of the robot are obtained by using information from agraph known as the roadmap [2 3] The learning phaseis the process of constructing a graph that represents themorphology of the given configuration space In this phaseit is important to construct the graph appropriately to obtainthe shortest path for the collision-free motion of the robotDevelopments have recently been reported on the optimalityof sampling-based algorithms [6ndash9] For example the PRMlowastalgorithm provides criteria for connecting edges that canguarantee asymptotic optimality in terms of the probabilityof finding path

An important factor to consider when constructing agraph for optimal motion planning is the node samplingstrategy used Sampling-based algorithms were originallydesigned with a random sampling of uniform distribution[2 10] However a few drawbacks have been discoveredin these uniform sampling strategies such as the narrowpassage problem and failure to capture the true connectivityof spaces of arbitrary shapes [9ndash11] Thus various samplingschemes such as the Voronoi diagram [11] Gaussian randomsampling [12] the bridge test [13] and visibility PRM [14]

HindawiComplexityVolume 2018 Article ID 9104720 23 pageshttpsdoiorg10115520189104720

2 Complexity

algorithms have been proposed to overcome drawbacks inthe uniform sampling method Nevertheless the effect ofsampling strategies on optimal motion planning remainsunclear The node sampling strategy that fully captures theshape of the free configuration space remains an importantproblem in optimal motion planning [9 10] In the queryphase of the roadmapmethod the shortest path is obtained asthe optimal path using graph informationThus constructingan appropriate graph is closely related to the problem ofoptimal motion planning

Another area highly relevant to our work here is researchon optimization techniques In recent years with grow-ing interest in machine learning and Artificial Intelligencetechniques of distributed optimization that can handlelarge amounts of data and high-dimensional variables haveemerged as an important issue In [15 16] distributed opti-mization techniques were considered formulti-agent systemsin complex networks Secure and resilient techniques [17ndash19] have also been developed to guarantee the stability ofdistributed algorithms in adversarial environments As highlyrelevant to ourwork here coveragemaximizationwas studiedusing methods of distributed optimization in applications ofsensor networks [20ndash22]

This paper proposes an algorithm to optimize theroadmap graph that can cover arbitrary morphologies ofthe free configuration space to maximize coverage Ourprevious study [23] considered an adaptation algorithm forgeometric graphs in an intuitive manner However in thisstudy we reanalyze that adaptive graph algorithm and presentits results in terms of the maximization of graph coverageWefound that the optimization algorithm updates the positionsof the nodes along the direction of expanding the region ofcovering until it reaches equilibrium where expansion andcontraction factors are balanced The results of a simulationto test the proposed optimization algorithm show that it canconstruct a suitable graph to describe arbitrary configurationspaces and the resulting graph can generate paths of shorterlength for various test queries

2 Problem Statement

The configuration space Q sub R119899 is the space of all possibleconfigurations of the system where n is the number ofdegrees of freedom (DOF) of the system and R119899 is n-dimensional Euclidean space [1] In the formulation of theconfiguration space the robot can be abstracted as a point q isinQ Free configuration is defined as one where the robot 119877(q)does not intersect with any obstacle in the workspaceThe setof all free configurations is defined as the free configurationspace Q119891119903119890119890 sub Q and it can be determined by checking forcollisions between obstacles and the robot in the workspaceEvery workspace obstacle 119882119874119894 is mapped as an obstacle inthe configuration space119876119874119894 = q isin Q | 119877(q) cap119882119874119894 = 120593 [2]Figure 1 shows the mapping relations between the workspaceand the configuration space for a two-link articulated robot

The problem of motion planning requires as solution afeasible path in the free configuration space for a specifiedinitial motion and a goal The path planning problem can thenbe defined as follows

Definition 1 ((path planning problem) [1]) Given a set ofcollision-free configurations Q119891119903119890119890 sub Q and initial and goalconfigurations q119894119899119894119905 q119892119900119886119897 isin Q119891119903119890119890 find a continuous curve120590 isin Σ = 119905 | 119905 [0 1] 997888rarr Q119891119903119890119890 where 120590(0) = q119894119899119894119905 and120590(1) = q119892119900119886119897

In this study we employ the roadmap method for pathplanning The roadmap represents the configuration spacetopologically by a network of one-dimensional (1D) curvesIt is defined as follows

Definition 2 ((roadmap) [2]) A union of 1D curves is aroadmap RM if for all q119894119899119894119905 and q119892119900119886119897 in Q119891119903119890119890 that can beconnected by a path the following properties hold

(1) Accessibility there exists a path from q119894119899119894119905 isin Q119891119903119890119890 tosome q1015840119894119899119894119905 isin RM

(2) Departability there exists a path from q1015840119892119900119886119897 isin RMto q119892119900119886119897 isin Q119891119903119890119890

(3) Connectivity there exists a path in RM between q1015840119894119899119894119905and q1015840119892119900119886119897

The result of the roadmap method is a graph G = (VE)consisting of a set of 119873 nodes V = q119894 isin Q119891119903119890119890 | forall119894 isin1 sdot sdot sdot 119873 and a set of edges E = 119890119894119895 = (q119894 q119895) | q119894 q119895 isin VEdge 119890119894119895 is created if the line segment between nodes q119894 andq119895 persists in Q119891119903119890119890 ie 119905q119894 + (1 minus 119905)q119895 isin Q119891119903119890119890 forall119905 isin [0 1]

To obtain a feasible path for the robotrsquos motion usinginformation from the roadmap graph the three propertiesin Definition 2 need to be met Furthermore for optimalplanning the best roadmap must be determined among a setof feasible roadmaps We now employ roadmap coverage asa performance index to assess the optimality of the roadmapgraph

To describe roadmap coverage the covering region of nodeq119894 for a certain neighbor radius 119903120578 is defined as follows

B119894 = B (q119894 119903120578) = q isin R119899 | 1003817100381710038171003817q minus q11989410038171003817100381710038172 le 1199031205782 q119894 isin V (1)

Let B = B1B2 sdot sdot sdot B119873 be the family of sets of B119894subsequently the roadmap coverage for neighbor radius 119903120578 isdefined as the union of B119894s as follows

Definition 3 (roadmap coverage) Roadmap coverage C forroadmap G is the region of the union of the family of setsB

C (G) = ⋃B = B1 ⋃B2 ⋃ sdot sdot sdot⋃B119873 (2)

Figure 2 shows an example of a roadmap and its coveragefor a configuration space

Let 120583(sdot) be the Lebesgue measure by the inclusionndashexclusion principle [25 26] the volume of roadmap coverage120583(C)can be represented as follows120583 (C (G)) = 120583 (⋃B) = 119873sum

119894=1(minus1)119894minus1 119878119894 (3)

where 119878119894 = sum1le1198951lt1198952sdotsdotsdotlt119895119894le119873 120583(B1198951 cap B1198952 cap sdot sdot sdot cap B119895119894)

Complexity 3

Collision motion

R(q)

Workspaceobstacle

Configuration space

Robotkinematics

Workspace Workspace

Workspace

Workspace

Workspace Workspace

QOi

free

WOi

Figure 1 Mapping relations between the workspace and the configuration space

Configuration space Coverage () = 1 cup 2 cup middot middot middot cup N

free

ir

2

i = (i r) = isin Rn minus i2 le

r

2 i isin |

Figure 2 Example of a roadmap and its coverage

According to the properties in Definition 2 the optimal-ity of the roadmap can be considered to be the capability offinding paths for an arbitrarily set initial point and goal inthe configuration space To satisfy this requirement roadmap

coverage should cover the entire free configuration space Asa quantitative representation of this performance index wedefine the optimal roadmap as the graph that maximizes thevolume of roadmap coverage as follows

4 Complexity

Definition 4 (optimal roadmap) The optimal roadmap Glowast isthe graph that maximizes the volume of roadmap coveragewithin the finite free configuration space

Glowast = maximizeV=11990211199022sdotsdotsdot 119902119873

120583 (C (G))subject to q119894 isin Q119891119903119890119890 119894 = 1 sdot sdot sdot 119873 (4)

The volume of roadmap coverage is primarily related tothe geometric structure of the graph It is complicated tocalculate especially in case the configuration space is highdimensional and the roadmap consists of a large number ofnodes Thus an alternative performance index that uses theupper or lower bound for the volume of roadmap coverage isneeded to render the optimization problem solvable

Let the sum of terms consisting of two or more intersec-tions be 119868(B) = sum119873

119894=2(minus1)119894119878119894 Therefore (3) can be written as120583 (C (G)) = 120583 (⋃B) = 119873sum119894=1

(minus1)119894minus1 119878119894= 1198781 + 119873sum

119894=2(minus1)119894minus1 119878119894 = 119873sum

119894=1120583 (B119894) minus 119868 (B) (5)

where sum119873119894=1 120583(B119894) = 119873 sdot Γ(1198992 + 1)minus1(radic120587 sdot 119903120578)119899 and Γ(119909) is

the gamma function The problem of designing the optimalroadmap can be transformed into aminimization problem asfollows

Glowast = minimizeV=11990211199022sdotsdotsdot 119902119873

119868 (B)subject to q119894 isin Q119891119903119890119890 119894 = 1 sdot sdot sdot 119873 (6)

Moreover the upper bound of 119868(B) which is the sum of theintersection terms can be expressed as follows [Section A1]

119868 (B) le 120581 sdot 1198782 = 120581 sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (7)

where 120581 = 2119873(119873 minus 1) sdot sum119873119894=2 (∐119894

119895=2 (119873119895 )) is a finite constant

that depends on the number of nodes NThis upper bound on 119868(B) can be applied to the following

suboptimal roadmap problem

Definition 5 (suboptimal roadmap) The suboptimal roadmapGlowast is the graph that minimizes the upper bound of 119868(B) in(7) such that


sum1le119894lt119895le119873

120583 (B119894 cap B119895)subject to q119894 isin Q119891119903119890119890 119894 = 1 sdot sdot sdot 119873 (8)

3 Coverage Maximization Algorithm

Let Φ(q119894q119895) be the volume of the region of intersectionbetween nodes 120583(B119894 cap B119895) By replacing the constraintsq119894 isin Q119891119903119890119890 with the collision detection function 119904(q119894) theoptimization problem in Definition 5 can be represented asfollows


sum1le119894lt119895le119873

Φ(q119894 q119895)subject to 119904 (q119894) = 0 119894 = 1 sdot sdot sdot 119873 (9)

where 119904(q119894) is the collision detection function defined as

119904 (q119894) = 1 (collision detected)0 (otherwise) (10)

The volume of the region of intersection between n-dimensional hyperspheres with respect to the neighborradius 119903120578 is

Φ(q119894 q119895) = 2 sdot int1199031205782

1198891198941198952

radic120587119899minus1

Γ (1 + (119899 minus 1) 2)radic(1199031205782)2 minus 1199052119899minus1119889119905 (0 le 119889119894119895 lt 119903120578)0 (119889119894119895 ge 119903120578) (11)

where 119889119894119895 = q119894 minus q1198952 See [Section A2] for detailsUsing the Lagrange multiplier 120582 the optimization prob-

lem in (9) can be transformed as follows [27]

Glowast = minimize H (12)

whereH = sum1le119894lt119895le119873Φ(q119894 q119895) + sum119873119894=1 120582119894119904(q119894)

The solution of this minimization problem is a nodeset that makes the gradient of H to zero ie nablaH =sum1le119894lt119895le119873nablaΦ(q119894 q119895) + sum119873

119894=1 120582119894nabla119904(q119894) = 0

Let x = [q119894] isin R119899119873 be the stacked vector of N nodesUsing the steepest descent method [27] the algorithm tominimize (9) can be described as follows

x [119896 + 1] = x [119896] minus 120576nablaH119896nablaH119896 = sum1le119894lt119895le119873

nablaΦ (q119894 [119896] q119895 [119896])+ 119873sum

119894=1120582119894 [119896] nabla119904 (q119894 [119896]) (13)

Complexity 5

Neighboring nodes

Neighbor radius

Node

r

i

Figure 3 Nodes neighboring q119894 within radius 119903120578 [23]where 119896 is the iteration index and 120576 gt 0 is the step size ofeach iteration By the penalty parameter 119888[119896] the Lagrangemultiplier 120582119894[119896] can be determined by the following updatingformula [27]120582119894 [119896 + 1] = 120582119894 [119896] + 119888 [119896] 119904 (q119894 [119896]) (14)

where the penalty parameter is chosen to satisfy 119888[119896 + 1] ge119888[119896]If the distance 119889119894119895 = q119894 minus q1198952 between nodes q119894 and

q119895 is greater than the neighbor radius 119903120578 the volume of theregion of intersection becomes zero ie Φ(q119894q119895) = 0 AlsonablaΦ(q119894 q119895) for node q119894 is valid only when q119895 is located within119903120578 Thereby as shown in Figure 3 we define the neighboringnode set located within the neighbor radius as

120578119894 = q119895 isin V | 10038171003817100381710038171003817q119895 minus q119894100381710038171003817100381710038172 le 119903120578 119895 = 119894 (15)

Furthermore using the decomposition method [28ndash30] theprocess in (13) can be transformed into a decentralized formthat is processed at each node as follows

q119894 [119896 + 1] = q119894 [119896] minus 120576 sumq119895[119896]isin120578119894

nablaΦ(q119894 [119896] q119895 [119896])minus 120576120582119894 [119896] nabla119904 (q119894 [119896]) (16)

nabla119904(q119894) in (16) is however not available because 119904(q119894)is discontinuous In such cases the gradient of 119904(q119894) canbe estimated stochastically through response surface method(RSM) [31ndash33] To apply the RSM algorithm we define thesensing node Δq isin R119899 as follows

Definition 6 (sensing node) A sensing node Δq is a pointevenly sampled by Poisson-disk sampling [34] on 120597B119903119878 =q isin R119899|q2 = 119903119904 which is the surface of an n-dimen-sional hypersphere of radius 119903119904 The sensing nodes detect the

deformation of the configuration space around each nodeThe set of sensing nodes S can be expressed as

S = Δq119897 isin 120597B119903119878 | 119897 = 1 sdot sdot sdot 119899119904 10038171003817100381710038171003817Δq119894 minus Δq119895100381710038171003817100381710038172gt 119903119901 for forall119894 119895 isin 1 sdot sdot sdot 119899119904 (17)

where 119903119901 and 119899119904 denote the sampling radius of the Poissondisk and the number of sensing nodes respectively

Figure 4(a) shows an example of sensing nodes sampledon the surface of a sphere and Figure 4(b) shows a node andits sensing nodes in 3D configuration space

LetM = [Δq119879119897 ] isin R119899119904times119899 be a matrix composed of sensingnodes and w119894 = [119904(q119894 + Δq119897)] isin R119899119904 be a stacked vectorrepresenting sensing information If the sensing nodes areselected symmetrically such that sum119899119904

119897=1 Δq119897 = 0 the estimatedgradient nabla119904(q119894) can be calculated by the RSM as followsnabla119904 (q119894) asymp nabla119904 (q119894) = nabla119904 (q119894) = M+w119894 (18)

where M+ = (M119879M)minus1M119879 is the pseudoinverse matrix ofM and M119879M is a nonsingular matrix See [Section A3] fordetailsΦ(q119894q119895) represents the interaction between two nodes q119894and q119895 and it is a function of the distance 119889119894119895 = q119894 minus q1198952The gradient of Φ(q119894 q119895) can be described as follows [35]nablaΦ (q119894 q119895) = 120597Φ (119889119894119895)120597119889119894119895 sdot q119894 minus q119895119889119894119895 (19)

Then it follows from (11) that120597Φ (119889119894119895)120597119889= minus2 radic1205874119899minus1Γ (1 + (119899 minus 1) 2)radic1199032120578 minus 1198892119894119895119899minus1 (0 le 119889119894119895 lt 119903120578)0 (119889119894119895 ge 119903120578)

(20)

6 Complexity

minus66

minus4

4

minus2

6

0

2 4

2

0 2

4

0

6

minus2 minus2minus4 minus4minus6 minus6

(a)

minus88

minus6

minus4

6

minus2

84

0

62

2

4

4

0 2

6

minus2 0

8

minus2minus4 minus4minus6 minus6minus8 minus8(b)

Figure 4 Example of sampled sensing nodes and a node in 3D configuration space (a) Sensing nodes sampled on 120597B119903119878 (119903119904 = 5) (b) A nodeand its sensing nodes

It is noteworthy that 120597Φ(119889119894119895)120597119889 le 0 We now set a non-negative value of 119892(119889119894119895) as

119892 (119889119894119895) = minus 1119889119894119895 sdot 120597Φ (119889119894119895)120597119889119894119895 (0 lt 119889119894119895 lt 119903120578)0 (otherwise) (21)

Subsequently the distributed optimization algorithm in (16)can be reformulated as follows

q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894

119892 (119889119894119895 [119896]) (q119894 [119896] minus q119895 [119896])minus 120576120582119894 [119896]M+w119894 [119896] (22)

In this algorithm the major variable influencing theamount of computation needed is the number of nodes NThe main computational tasks of the optimization processare the construction of the Laplacian matrix and the com-putation of the sensing vector In the original optimizationprocess in (13) computational complexities according to thenumber of nodes N are estimated as quadratic 119874(1198732) forthe Laplacian matrix and linear 119874(119873) for the sensing vectorrespectively However as this algorithm can be decomposedas shown in (16) if a many-core processor such as a General-Purpose computing on Graphics Processing Unit (GPGPU)is available (22) can be computed at each node in a separateprocessor in parallel Thus if the optimization algorithm canbe executed by using parallel computation the computationalcomplexity of the Laplacian matrix and the configurationof the sensing vector are linear 119874(119873) and constant 119874(119888)respectively

The algorithm in (22) is an iterative process that canconverge to a suboptimal roadmap by detecting the boundaryof the state of collision of the configuration space throughthe sensing vector w119894 Even if the morphology of theconfiguration space changes the structure of the roadmapgraph can adapt to it accordingly because the optimizationprocess is continuously performed using the sensing node Inthe general optimization algorithm the Lagrange multiplier120582119894[119896] is updated at every iteration This is suitable in fixedconfiguration spaces but it is difficult to apply to changeablespaces because if the space changes the updated 120582119894[119896]becomes useless and needs to be recalculated to obtainthe initial conditions We set the Lagrange multiplier to aconstant value 120582 as an iteration gain Further we set the samevalue for each node such that 1205821 = 1205822 = sdot sdot sdot = 120582119873 = 120582The results of a simulation show that the algorithmworkswellwith a fixed 120582 even with a changing configuration space

4 Equilibrium State Analysis

This section analyzes the states of equilibrium of the algo-rithm in (22) For the equilibrium analysis the differentialequation for the vector x = [q119894] isin R119899119873 should be considered

Let the vector w = [w119894] isin R119899119904119873 be the stacked vector ofsensing vector w119894 at each node and let the Laplacian matrixL = [119871 119894119895] isin R119873times119873 for 119892(119889119894119895) ge 0 be as follows

L = [119871 119894119895] 119871 119894119895 = minus119892 (119889119894119895) 119894 = 119895

119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (23)

Complexity 7

(a) (b)

Figure 5 Examples of states of equilibrium The yellow region shows the coverage of the roadmap (a) An ill-structured roadmap (b) Aproperly constructed roadmap

With the Lagrange multipliers 120582119894 set to 1205821 = 1205822 = sdot sdot sdot = 120582119873 =120582 the decentralized differential equation shown in (22) canbe integrated for x isin R119899119873 as follows [Section A4]

x [119896 + 1] = x [119896] + 120576 (L [119896] otimes I119899) x [119896]minus 120576120582 (I119873 otimesM+)w [119896]= (I119899119873 + 120576 (L [119896] otimes I119899)) x [119896]minus 120576120582 (I119873 otimesM+)w [119896] (24)

where otimes is the Kronecker product operator and I119896 isin 119877119896times119896 is ak-dimensional identity matrix

Let xlowast be the whole-state variable in equilibrium and letwlowast and Llowast be the entire sensing vector and the Laplacianmatrix at that time respectively Subsequently the differentialequation at equilibrium is

xlowast = (I119899119873 + 120576 (Llowast otimes I119899)) xlowast minus 120576120582 (I119873 otimesM+)wlowast= xlowast + 120576 (Llowast otimes I119899) xlowast minus 120576120582 (I119873 otimesM+)wlowast (25)

In (25) the condition that renders the whole-state vectorx in equilibrium is120576 (Llowast otimes I119899) xlowast minus 120576120582 (I119873 otimesM+)wlowast = 0 (26)

The cases of states of equilibrium that can meet such acondition can be summarized as follows

Remark 7 (equilibrium conditions of suboptimal roadmapproblem) The conditions of equilibrium states for the sub-optimal roadmap algorithm are classified into three cases asfollows

Case 1 (Llowast = 0wlowast = 0) All nodes and their correspondingsensing nodes are in Q119891119903119890119890 Furthermore no neighbor nodeexists because the neighbor radius 119903120578 is small for 120583(Q119891119903119890119890)

such that 120583(Q119891119903119890119890) gt 119873 sdot 120583(B(q119894 119903120578)) Figure 5(a) shows anexample of this case

Case 2 (Llowast = 0 xlowast isin 119873119906119897119897(Llowast otimes I119899)wlowast = 0) In thiscase x is a null-space element of the matrix Llowast otimes I119899 and allsensing nodes are in Q119891119903119890119890 If the graph is formed by a singleconnected component the null-space elements are x = [q119894]where q1 = q2 = sdot sdot sdot = q119873 [36] Such states are rare indynamical processes Thus herein we do not consider thistype of equilibrium

Case 3 (Llowast = 0 xlowast notin 119873119906119897119897(Llowast otimes I119899)wlowast = 0) The neighbornodes and information regarding the sensing nodes exist Inthis case the equilibrium is a point and this is the desirablesolution to the suboptimal roadmap problem Figure 5(b)shows an example of this case In this state the expandingfactor between the neighbor nodes and the factor of sensingnodes that detect the boundary of collision are in balance asfollows (Llowast otimes I119899) xlowast = 120582 (I119873 otimesM+)wlowast (27)

The maximum rank of the Laplacian matrix is nminus1 [37]Although information concerning the Laplacian matrix Llowastand sensing vector wlowast in equilibrium state is given the statevector xlowast cannot be calculated using (27) However if thegraph is a single connected component such as 119903119886119899119896(Llowast) =119899 minus 1 [36 37] the state vector xlowast can be obtained with someadditional constraints

If a constraint such assum119873119894=1 q119894 = 0 is added the state vector

in equilibrium can be calculated as follows

xlowast = ([[ Llowast1119873 sdot 1119879119873]] otimes I119899)+ [120582 (I119873 otimesM+)wlowast

0] (28)

where the symbol ldquo+rdquo indicates the pseudoinverse matrix and1119873 is an N-dimensional vector

8 Complexity

2 1-1 1

rS

Figure 6 Structure of roadmap for Example 8

We now verify this result through a simple example ofa roadmap with two nodes in a 1D configuration space asshown in Figure 6

Example 8 (two-node roadmap in 1D space) Obtain theequilibrium point 119909lowast = [ 119909lowast1

119909lowast2] for Q119891119903119890119890 = 119909 isin R | minus1 le119909 le 1 119903119904 = 06 120582 = 1 Llowast = [ 1 minus1minus1 1 ] and wlowast = [1 0 0 1]119879

The sensing node matrix is M = [ 06minus06 ] and its pseudo-

matrix isM+ = [08333 minus08333]The configuration space issymmetric to the origin Thus a constraint on the center ofmass can be added such as (1199091+1199092)2 = 0 Subsequently theequilibrium point can be calculated using (28) as follows

xlowast = [[[1 minus1minus1 112 12]]]

+

sdot [[[[[[[[([1 00 1] otimes [08333 minus08333])[[[[[[

1001]]]]]]0]]]]]]]]= [ 04167minus04167]

(29)

Hence the roadmap at equilibrium is as shown inFigure 7

Figure 8 shows the phase portrait of the optimizationprocess for various initial conditions As shown in the phaseportrait the states of x converge for all initial conditionsto xlowast = [04167 minus04167]119879 or xlowast = [minus04167 04167]119879as calculated in (29) In case of convergence to xlowast =[minus04167 04167]119879 the sensing vector is wlowast = [0 1 1 0]119879

Figure 9 shows the phase portrait of example 1 for 119903120578 =06 where the neighbor radius is small for the volume of theconfiguration space In this case the equilibrium conditionof the optimization process is the first case of Remark 7 Thestate of equilibrium is not a point but a region here as shownby the two symmetrical triangles in Figure 9

5 Regulation of Internal Repulsion

From the differential equation in (24) the types of equilib-rium can be summarized into three cases We also confirmedthat the equilibrium of Case 3 such that

Llowast = 0xlowast notin 119873119906119897119897 (Llowast otimes I119899) wlowast = 0 (30)

is the desirable solution to the suboptimal roadmap problemTo construct the appropriate roadmap the Laplacian matrixmust not be a zero matrix To investigate this issue theinternal repulsion is defined as the quantitative informationof the Laplacian matrix

Definition 9 (internal repulsion) The internal repulsion 119875 isthe sum of the repulsion factors between neighbor nodes inthe graph It can be defined as the entry-wise 1-norm of theLaplacian matrix as follows119875 = 12 L1 = 12 vec (L)1 = 12 119873sum

119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 (31)

In the Laplacian matrix the diagonal element is the sum ofthe row elements except itself such that |119871 119894119894| = sum119873

119894=1119894 =119895 |119871 119894119895|Considering (23) the internal repulsion can be representedas follows119875 = 12 119873sum

119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 = 12 119873sum

119894=12 119873sum119895=1119894 =119895

10038161003816100381610038161003816119892 (119889119894119895)10038161003816100381610038161003816= 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119889119894119895) (32)

Equilibrium in Case 1 can occur when the neighborradius is small in comparison with the volume of the freeconfiguration space such that 120583(Q119891119903119890119890) gt 119873 sdot 120583(B(q119894 119903120578))Thus whether the Laplacian matrix is a zero matrix is inti-mately related to the volume of the free configuration space120583(Q119891119903119890119890) and neighbor radius 119903120578 Therefore to determine theproperties of convergence of the algorithm we observe thesteady-state value of internal repulsionwith respect to variousneighbor radii

Complexity 9

05833 08334 05833

-04167 04167

2 1

Figure 7 State of equilibrium for Example 8

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

lowast = [minus0416704167]

lowast = [ 04167minus04167

]

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a)

minus1

minus05

0

05

1

minus1minus05

005

1x1

x2

005

115

225

335

cost

(b)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x1

(c)

05

1

15

2

25

3

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x1(d)

Figure 8 Phase portrait of the optimization process of Example 8 (a) State trajectories for each initial condition (b) Cost graph of functionH (c) Contour graph of the cost function (d) Cost graph

Figure 10 shows the maximum and minimum values ofinternal repulsion in the steady state for various neighborradii Figure 11 shows the structure of the roadmap in thesteady state for some cases In the internal repulsion graphthe maximum and minimum values are identical when theneighbor radius is smaller than 57 In this case the internalrepulsion and the nodes remain in a certain state whichmeans that the structure and the state of the roadmap con-verge to equilibrium However when the internal repulsionconverges to zero in the case of a neighbor radius smaller

than 40 this means that the roadmap does not encompassthe entire free configuration space as shown in Figures 11(a)and 11(b) When the neighbor radius is 47 as shown inFigure 11(c) the structure of the roadmap encompasses theentire free configuration space well because the state of theroadmap converges with the appropriate internal repulsionsuch as in the equilibrium of Case 3 When the neighborradius is larger than 57 the maximum and minimum valuesare different This means that the state of the nodes andthe internal repulsion oscillate without converging In this

10 Complexity

1

minus1minus08minus06minus04minus02

002040608

1

minus1 04

02

06

08

minus02

minus08

minus04

minus06 0

(a)


002040608

1

x2

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1

x1(b)


002040608

1

x2

0 1minus1 02

06

08

04

minus02

minus06

minus08

minus04

x1(c)

minus1

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1

minus08minus06minus04minus02

002040608

1

x1

x2

0020406081121416182

(d)

Figure 9 Phase portrait of the optimization process of Example 8 for 119903120578 = 06 (a) State trajectories for each initial condition (b) Statetrajectories and contour graph (c) Contour graph (d) Cost graph of function H

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

57 59 61 63 65 67 69

Neighbor radius vs Internal repulsion

minimummaximum

Converge toan equilibrium

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55

Neighbor radius

Inte

rnal

repu

lsion

Not converge

Converged toa state of zero(not cover Ｌ)

Figure 10 Graph of internal repulsion with respect to neighbor radius 119903120578

Complexity 11

(a) (b) (c) (d)

Figure 11 States of roadmap after 100 iterations (a) 119903120578 = 10 (no action) (b) 119903120578 = 30 (ill-structured graph) (c) 119903120578 = 47 (well-structured graph)(d) 119903120578 = 60 (does not converge)case the algorithm cannot properly construct the roadmapbecause the state does not converge to equilibrium as shownin Figure 11(d)

The deformation of the configuration space can inducechanges in the volume of the free configuration space Whenthe volume of the free configuration space changes theroadmap may no longer be constructed properly because theneighbor radius may be large or small relative to the changedvolume of the free configuration space Thus the parameterof the neighbor radius should be adjusted with respect tothe volume of the free configuration space Generally thevolume of the free configuration space is difficult to calculateHowever the internal repulsion is highly related to it and iscomputable by executing the algorithm With the regulationof internal repulsion by a desirable value the roadmapcan be constructed stably regardless of the volume of thefree configuration space To regulate internal repulsion thecontroller that adjusts the neighbor radius should satisfy thefollowing conditions

Remark 10 (conditions for internal repulsion regulation) Ifthe repulsion function and the regulation controller meetthe following conditions for L = 0 the internal repulsionregulation can be achieved See [Section A5] for proof

(1) Repulsion function condition120597119892 (119903120578 120588119894119895)120597119903120578 gt 0 (exist120588119894119895 120588119894119895 lt 119903120578) (33)

where 119903120578 is added to the repulsion function as avariable because it is the control value in internalrepulsion regulation and 120588119894119895 is the expectation ofneighbor distance ie 120588119894119895 = E[119889119894119895]

(2) Regulation controller condition119903120578 = 119866119868119877119877 int119890119875119889119905 + 119903120578 (1199050) (119890119875 = 119875119889 minus 119875120588) (34)

where 119866119868119877119877 gt 0 is controller gain 119875119889 is thedesired internal repulsion value and 119875120588 is the internalrepulsion by the expectation of neighbor distance ie119875120588 = sum119873

119894=1 sum119873119895=1119894 =119895 119892(119903120578E[119889119894119895])

In the suboptimal roadmap algorithm the gradient of therepulsion function satisfies Condition (1) because it is calcu-lated from (21) as follows120597119892120597119903120578

= 4119889119894119895 radic1205874119899minus1Γ (1 + (119899 minus 1) 2)119903120578radic1199032120578 minus 1198892119894119895119899minus3 (0 lt 119889119894119895 lt 119903120578)0 otherwise

(35)

Thus the internal repulsion can be regulated to the desiredvalue using the controller of Condition (2) Such a controllerin (34) can be implemented as a discrete-time system asfollows119903120578 [119896] = 119866119868119877119877sdot 119896sum

119897=1198960120576 sdot (119875119889 minus 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119903120578 [119897] 120588119894119895 [119897]))+ 119903120578 [1198960] (36)

where 120576 is the sampling time or step size of the discrete-time system and 120588119894119895[119896] is the estimation of 120588119894119895 that can becalculated as the average of 119889119894119895 for 119879119877 intervals as 120588119894119895[119896] =(1(119879119877 + 1))sum119896

119895=119896minus119879119877 119889119894119895[119895]Figure 12 shows the efficacy of internal repulsion regu-

lation in the case of a sudden change in free configurationspace At the initial condition no obstacle exists and thefree configuration space ratio 120583(Q119891119903119890119890)120583(Q) is 100 Afterthe process reaches the state of equilibrium some obstaclesare added so that the volume of the free configuration spaceis reduced to 693 The graphs of the internal repulsionfor this situation are depicted in Figure 12 with respect tothe time series The green line represents the fixed neighborradius and the blue line corresponds to the internal repulsionregulation As shown in the graph if no obstacles exist inthe configuration space both roadmaps converge to a similarstructure However after the addition of obstacles into theconfiguration space the roadmap with the fixed neighborradius cannot converge and exhibits unstable oscillation asdepicted by the green lineHowever the blue line graph shows

12 Complexity

0

1000

2000

3000

4000

5000

6000

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201

Energy regularizationFixed neighbor radius

Configuration space deformation(free)() 100 rarr 693

Figure 12 Graphs of internal repulsion for a change in the configuration space with a fixed neighbor radius (119903120578 = 50) and with internalrepulsion regulation (119875119889 = 700)

An arbitrary shape ofconfiguration space

w[k]

Suboptimal Roadmap Algorithm

Roadmap

+

minus

Internal repulsion(Entrywise 1-norm ofthe Laplacian matrix)

DesiredInternal repulsion

Error Controller

Internal Repulsion Regulation

Edgeconstruction

x[k]

x[k + 1]+

minus V

E

G = (V E)

Node

Edge

Unit delay

P =1

21

zminus1

nN + [k] otimes n

r

[k + 1] = (nN + [k ] otimes n) middot [k] minus (N otimes +) middot [k]

(N otimes+)

PdeP

GIRR int eP dt

Figure 13 Overall structure of the suboptimal roadmap algorithm with internal repulsion regulation

that the roadmap converges to equilibrium by regulating theinternal repulsion uniformly regardless of the volume of thefree configuration space

Figure 13 shows an overall diagram of the optimizationalgorithm with internal repulsion regulation This systemfeatures two types of differential equations One is the differ-ential equation used tomaximize coverage that can construct

an appropriate graph against deformations in the config-uration space and the other is that for internal repulsionregulation of the roadmap regardless of the volume of thefree configuration space With these differential equationsthe suboptimal roadmap algorithm can construct the properroadmap graph that covers arbitrary shapes of the freeconfiguration space

Complexity 13

Initialize nodes

Constructing sensing nodes(Poisson disk sampling on n-sphere)

Creating distributed threads

Start

Resume threads

Synchronize

Terminate

Quit

Destroying distributed threads

Parallel Processing

Processing for neighboring nodes(construction of Laplacian matrix)

Processing for sensing vector(Collision checking on sensing nodes)

Update node

Update edge

N

Y

Timer event

Internal Repulsion Regulation(Computing Equation (36))

At each node

(Computing Equation (22) for i)

(Random sampling on free)

i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]

r[k] = GIRR middotksum

l=k0

middot(Pd minus Nsumi=1

Nsumj=1i =j g(r[l] ij[l])) + r[k0]

i[k] = sum[k]isin

g(dij[k])( )i[k] minus j[k]

Figure 14 Flowchart for processing the suboptimal roadmap algorithm with internal repulsion regulation

The suboptimal roadmap algorithm can be optimized bydividing its complex procedures using parallel computingtechniques such as multithreading or GPU processing Thus(22) is computed concurrently at each node Figure 14 showsthe flowchart for processing the suboptimal roadmap algo-rithmwith internal repulsion regulationThe four proceduresin the red border are performed in parallel at each node

6 Case Study

To verify the effectiveness of the proposed method weapplied the suboptimal roadmap algorithm to a robot systemin a car manufacturing assembly line The target robotwas the MPX3500 a painting robot system of the YaskawaElectric Corporation (YEC) [24] As shown in Figure 15 theMPX3500 robot had six axes the S L and U axes renderedtranslation motions of the end effector The other axesmdashRB and Tmdashprovided orientation motions In conventionalindustrial robots these axes are located around the endeffector In this application only three axesmdashthe S L and

U axesmdashare considered as the orientation-related motions ofthe end effector do not affect motion related to collisionsHence the configuration space of the MPX3500 was a 3Dspace and the constructed roadmap could be visualized witha geometrical representation

The first step in applying the motion-planning algorithmis to create the collision-detection function 119904(q) We imple-mented it using the oriented bounding box (OBB) methodwhich yields the best efficiency in terms of both accuracy andcomputation time [38] The suboptimal roadmap algorithmwas applied to plan the motion of the MPX3500 robotin the automobile manufacturing line Figure 16 shows thesimulation environment where a model of a car and someobstacles were placed To observe the performance of theconstructed roadmap with respect to the number of nodesN we varied N from 50 to 300 and set the step size and theLagrange multiplier to 120576 = 001 and 120582 = 25 respectively Thenumber of sensing nodes and the radius of the sensing nodewere set to 119899119904 = 100 and 119903119904 = 5 respectively and the desiredinternal repulsion was set to 119875119889 = 100

14 Complexity

S

L

U

T

B

R

(a) (b)

Figure 15 YEC-MPX3500 painting robot system [24] (a) MPX3500 robot (b) Scene of its operation at an automobile painting line

Figure 16 Workspace of an automobile manufacturing lineThe car model and some obstacles are placed in the simulation environment

Figure 17 shows the state of the roadmap for the itera-tions of the suboptimal roadmap algorithm for 150 nodesFigure 17(a) displays the initial state of the roadmap in which150 nodes were scattered randomly in the configurationspace Figures 17(b) and 17(c) show the states of the roadmapafter the first and the fifth iterations respectively These statesare the initial steps of the execution of the algorithm Figures17(d) 17(e) and 17(f) show the states of the roadmap afterthe 25th 50th and 100th iterations respectively After 25iterations no considerable changes were observed in the stateof the roadmap which indicates that the iteration processmight have arrived at a minimum state

Figure 18 shows roadmaps of the suboptimal roadmapalgorithm with respect to the number of nodes after 100iterations As shown in the figures the appearances of theroadmap graphs are similar However the density of nodesis different with respect to the number of nodes Althoughthe quality of the results of planning depends on the densityof nodes near-optimal solutions were obtained even fora low density of nodes because the nodes were dispersedevenly

Figures 19 and 20 show examples of the results of motionplanning using the optimal roadmap PRMlowast and RRTalgorithms in the test environment Figure 19(a) shows theinitial motion and (b) shows the goal motion of the robot

(c) and (d) present the compared paths obtained by thethree methods in the configuration space Figure 20 showsthe three trajectories of the robotrsquos motion according to theplanned path in Figure 19 in the workspace As shown inthe figures the paths obtained by the suboptimal roadmapmethod were the shortest for the two workspaces thoseobtained by the PRMlowast algorithm were the next shortest andthe RRT algorithm offered suboptimal results such as pathsinvolving detours as confirmed in Figures 19(c) and 19(d)

Figure 21 shows comparison graphs of the results ofmotion planning using different algorithmsmdashthe suboptimalroadmap PRMlowast and RRTmdashwith respect to the number ofnodes N Using roadmaps with varying number of nodeseach calculation was performed 10 times to plan for 100pairs of motions of arbitrary initial points and goals For allroadmaps the results of the suboptimal roadmap algorithmyielded the lowest value of accumulated length of the plannedpath This indicates that the paths obtained using the pro-posedmethod were closest to the optimal results for the givenmotion-planning problem Figure 22 shows the graph of thecumulative length of the paths obtained with the suboptimalroadmap and the PRMlowast algorithm with respect to thenumber of nodes As shown in the figure the graph of thesuboptimal roadmap algorithm is always lower than that ofthe PRMlowastThis indicates that better motion-planning results

Complexity 15

(a) (b) (c)

(d) (e) (f)

Figure 17Operating procedures of the suboptimal roadmap algorithm in the configuration space (150 nodes) (a) Initial state (b) 1st iteration(c) 5th iteration (d) 25th iteration (e) 50th iteration (f) 100th iteration

(a) (b) (c)

(d) (e) (f)

Figure 18 Roadmaps constructed by the suboptimal roadmap algorithm with respect to the number of nodes after 100 iterations (a)N = 50(b) N = 100 (c) N = 150 (d) N = 200 (e) N = 250 (f) N = 300

can be obtained with the suboptimal roadmap algorithm witha small number of nodes Moreover the ratios of decrementof the suboptimal roadmap graphs were smaller than thoseof PRMlowast This means that the motion-planning perfor-mance of the suboptimal roadmap algorithm was less sen-sitive to the number of nodes than current sampling-basedmethods

7 Conclusion

This paper considered the problem of constructing a properroadmap graph that fully encompasses the arbitrary mor-phologies of the free configuration space To this end thecoverage of the graph was defined as a performance index onthe optimality of a graph constructed by the sampling-based

16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)

Figure 19 Comparison of the results of motion planning in the test environment (a) Initial configuration (b) Goal configuration (c) (d)Motion paths obtained by the proposed method (red) PRMlowast (magenta) and RRT (blue) algorithms in the configuration space

algorithm We then proposed an optimization algorithm thatcan maximize graph coverage in the configuration spaceBecause of the computational complexity of coverage in prac-tice the proposed algorithm was designed using the conceptof the suboptimal roadmap that minimizes the upper boundof the volume of intersection of the graph-covering regionEquilibrium conditions for the optimization algorithm werederived from the iteration equation Among the resultswe found the equilibrium condition that can construct anappropriate roadmap in an arbitrary configuration spaceFurthermore a method for regulating internal repulsionwas proposed to utilize the suboptimal roadmap algorithmregardless of changes to the volume of the free configurationspace The convergence of the algorithm was found to behighly relevant to the volume of the free configuration spaceaccompanied by the number of nodes and the neighbor-ing radius Thus to represent these relations quantitativelyinternal repulsion was defined as the sum of repulsionfactors between neighboring nodes in the graph Furtherwe proposed an integrator-type controller that adjusts theneighboring radius to regulate internal repulsion according tothe desired value The feasibility of the proposed method wasconfirmed through a case study involving a robotWe appliedthe method to an industrial robot used in car manufacturingassembly lines The results of the case study confirmed thatthe roadmap graph obtained by the proposed algorithm canproducemore optimal results for paths of motion of the robotin the simulation environment

Appendix

A

A1 Upper Bound of Intersection Volume 119868(B)Theorem A1 (upper bound of intersection volume) In afamily of sets B = B1B2 sdot sdot sdot B119873 composed of N setsthe upper bound of the intersection volume 119868(B) can berepresented by the multiplication of the constant 120581 and the sumof the intersection volume of two sets as follows119868 (B) le 120581 sdot ( sum

1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)



related to number of sets119873

Proof Let the sum of intersection volumes of 119894 elements inBbe 119878119894 = sum

1le1198951lt1198952sdotsdotsdotlt119895119894le119873120583 (B1198951 cap B1198952 cap sdot sdot sdot cap B119895119894) (A2)

subsequently the intersection volume 119868(B) = sum119873119894=2(minus1)119894119878119894

meets the following condition119868 (B) = 119873sum119894=2

(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)

Figure 20 Trajectories of the robotrsquos motion in the workspace with respect to the planning results of Figure 19 (a) Proposed method (b)PRMlowast (c) RRT

Because 119878119894+1 le ( 119873119894+1 ) sdot 119878119894 by Lemma A2 the following

inequality is met for 119878119894119878119894 le (1198732)minus1 ( 119894∐119895=2

(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)

further the following inequality is also satisfied forsum119873119894=2 119878119894 (ge119868(B))

119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2

(119873119895 )) sdot 1198782 + sdot sdot sdot + (1198732)minus1( 119873∐119895=2

(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2

(119873119895) sdot 1198782= 2119873 (119873 minus 1)sdot 119873sum119894=2

( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10

Proposed methodPRMlowast

RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)

Figure 21 Results of comparison of path qualities obtained using the proposed method PRMlowast and RRT algorithms with respect to thenumber of nodes (a) N = 50 (b) N = 100 (c) N = 150 (d) N = 200 (e) N = 250 (f) N = 300

0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N

Figure 22 Results of evaluation of the total cumulative path of the proposed method and PRMlowast algorithm with respect to the number ofnodes in the test environment

Complexity 19

Hence the intersection volume 119868(B) satisfies the inequalityof (A1)

Lemma A2 119878119894+1 meets the following inequality for 119878119894119878119894+1 le ( 119873119894 + 1) sdot 119878119894 (A6)

Proof Let the k-th intersection set in 119878119894 be C119894(119896) = B1198951(119896) capB1198952(119896) cap sdot sdot sdot cap B119895119894(119896) and its volume be 120583(C119894(119896)) therefore 119878119894can be expressed as follows119878119894 = sum

1le1198951lt1198952sdotsdotsdotlt119895119894le119873120583 (B1198951 cap B1198952 cap sdot sdot sdot cap B119895119894)

= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)

In 119878119894+1 as C119894+1(119896) = C119894(119897119896) cap B119895119894+1(119896) for 119897119896 isin 1 sdot sdot sdot (119873119894 )C119894+1(119896) sube C119894(119897119896) can be represented as120583 (C119894+1 (119896)) = 120572119894+1 (119896) sdot 120583 (C119894 (119897119896))

for 0 le 120572119894+1 (119896) le 1 (A8)

Thus 119878119894+1 can be expressed by the sum of ( 119873119894+1 ) terms of 120572119894+1 sdot120583(C119894) as follows

119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)

Let 120573119894+1(119897) be the sum of 120572119894+1(119896) that is applied to the same120583(C119894(119897)) therefore 119878119894+1 can be reformulated by the sum of(119873119894 ) terms of 120573119894+1 sdot 120583(C119894) as follows119878119894+1 = ( 119873

119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )

119896=1 120572119894+1(119896) = sum(119873119894 )119896=1 120573119894+1(119896)119878119894+1 can be represented by the inner product of two

vectors 120573119894+1 = [120573119894+1(119896)] isin R(119873119894 ) and 120583119894 = [120583(C119894(119896))] isin R(119873119894 )

as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)

Because 120573119894+1(119896) 120583(C119894(119896)) ge 0 119878119894+1 = 120573119879119894+1 sdot 120583119894 le 120573119894+11 sdot1205831198941 is satisfied by the CauchyndashSchwarz inequality 1205831198941 =sum(119873119894 )119896=1 120583(C119894(119896)) = 119878119894 and

1

2Φ(i j)

j

r

2

2

i i minus j = 2

Figure 23 Two spheres and their region of intersection

1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) le ( 119873119894 + 1) (∵ 0 le 120572119894+1 (119896) le 1) (A12)Hence the inequality in (A6) is satisfied as follows

119878119894+1 = 120573119879119894+1 sdot 120583119894 le 1003817100381710038171003817120573119894+110038171003817100381710038171 sdot 100381710038171003817100381712058311989410038171003817100381710038171 le ( 119873119894 + 1) sdot 119878119894 (A13)

A2 Volume of Region of Intersection between 119899-DimensionalHyperspheres [39] The volume of the n-dimensional hyper-sphere can be calculated by integrating the volume of an nminus1-dimensional hypersphere along a certain axis As shown inFigure 23 the region of intersection between spheres can becalculated by integrating the volume of the nminus1-dimensionalsphere from 1198891198941198952 (119889119894119895 = q119894 minus q1198952) to 1199031205782 along a certainaxis The volume of the nminus1-dimensional hypersphere ofradius 119903 is 119881119899minus1 (119903) = radic120587119899minus1

Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)

where Γ(119909) is the gamma function [40]To obtain the volume of the n-dimensional hypersphere

of radius 1199031205782 the nminus1 hypersphere is integrated for radius119903 = radic(1199031205782)2 minus 1199052 on minus1199031205782 le 119905 le 1199031205782 [39] Thus theintegral of the yellow region in Figure 23 can be calculatedby integrating the volume of the nminus1 sphere as followsint119881119899minus1 (119903) 119889119903

= int1199031205782

1198891198941198952


Γ (1 + (119899 minus 1) 2)radic(1199031205782 )2 minus 1199052119899minus1119889119905 (A15)

20 Complexity

In Figure 23 the yellow area calculated by the inte-gration is the half-region of the area of intersection thusthe volume of intersection is twice the calculated volume

Further 119889119894119895 ge 0 and Φ(q119894 q119895) = 0 for 119889119894119895 ge 119903120578 thusthe volume of the region of intersection between the n-dimensional hyperspheres can be summarized as follows

Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952


Γ (1 + (119899 minus 1) 2)radic(1199031205782 )2 minus 1199052119899minus1119889119905 (0 le 119889119894119895 lt 119903120578)0 (119889119894119895 ge 119903120578) (A16)

A3 Gradient Estimation of the Collision-Detection FunctionFunction 119904(q) checks for collisions in the workspace forconfiguration q isin Q As 119904(q) is a discontinuous functionits gradient cannot be calculated The gradient of 119904(q) canbe estimated stochastically using the response surface method(RSM) [31ndash33]

In the RSM for q119894 isin Q 120593 isin R119899 119904(q) is assumed to be alinear model in the neighborhood of q119894 as follows119904 (q) = 1205930 + 120593119879 (q minus q119894) (A17)

The gradient of the function above which is an estimatedgradient of 119904(q) at q = q119894 isnabla119904 (q)10038161003816100381610038161003816q=q119894 = nabla119904 (q119894) = 120593 (A18)

In the RSM algorithm nabla119904(q119894) can be estimated using pdesign points q119898 (119898 = 1 sdot sdot sdot 119901) around node q119894 here the119899119878 (gt 119899) sensing nodes of q119894 correspond to the design pointsie q119898 = q119894 + Δq119898 (119898 = 1 sdot sdot sdot 119899119878) By stacking the 119904(q)information of 119899119878 design points (A17) can be represented inmatrix form as follows

[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[

1205930 + 120593119879 (q119894 + Δq1 minus q119894)1205930 + 120593119879 (q119894 + Δq2 minus q119894)1205930 + 120593119879 (q119894 + Δq119899119904 minus q119894)]]]]]]]]

= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)

Using the sensing vector w119894 = [119904(q119894 +Δq119898)] isin R119899119904 at nodeq119894 the parameter 119868 = [ 1205930

] isin R119899+1 is estimated by the least-

squares algorithm as follows [32]

119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)

whereM = [Δq119879119898] isin R119899119904times119899 andM119868 = [1119899119878 M] isin R119899119878times(119899+1)

The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879

0 M119879M] therefore (A21)

becomes

119868 = [1205930] = [119899119878 0119879

0 M119879M]minus1 [1 sdot sdot sdot 1

M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)

Hence the estimated gradient of 119904(q) is = (M119879M)minus1M119879w119894 (A23)

A4 Derivation of the Differential Equation for the EntireOptimization Variable x isin R119899119873 Let the Lagrange multipliers120582119894 be 1205821 = 1205822 = sdot sdot sdot = 120582119873 = 120582 then the decentralizedoptimization algorithm can be formulated as follows

q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894

119892 (119889119894119895 [119896]) (q119894 [119896] minus q119895 [119896])minus 120576120582M+w119894 [119896] (A24)

where 119889119894119895 is the distance between nodes q119894 q119895 120578119894 is the set ofneighbors of q119894 and 119892(119889119894119895) is a function defined in (21)

Complexity 21

Because 119892(119889119894119895) = 0 for q119895 notin 120578119894 (A24) becomes

q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895

119892 (119889119894119895 [119896]) (q119894 [119896] minus q119895 [119896])minus 120576120582M+w119894 [119896] = q119894 [119896]+ 120576(( 119873sum119895=1119894 =119895

119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)

Let the row vector L119894 be

L119894 = [119871 119894119895] isin R1times119873 119871 119894119895 = minus119892 (119889119894119895) 119894 = 119895

119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)

(A25) can be represented for x = [q119894] isin R119899119873 as follows

q119894 [119896 + 1] = q119894 [119896] + 120576 (L119894 [119896] otimes I119899) x [119896]minus 120576120582M+w119894 [119896] (A27)

where otimes denotes the Kronecker product operatorThe equations of iteration performed at every node can

be integrated such that

[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]

]]]]]]] otimes I119899) x [119896]minus 120576120582 (I119873 otimesM+)[[[[[[[

w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)

Let w = [w119894] isin R119899119904119873 be the stacked vector of N sensingvectorsw119894 then using the Laplacian matrix L = [L119894] isin R119873times119873

(A28) can be reformulated for the entire variable x isin R119899119873 asfollows

x [119896 + 1] = x [119896] + 120576 (L [119896] otimes I119899) x [119896]minus 120576120582 (I119873 otimesM+)w [119896]= (119868119899119873 + 120576L [119896] otimes I119899) x [119896]minus 120576120582 (I119873 otimesM+)w [119896] (A29)

A5 Proof of Convergence For Internal Repulsion RegulationThis section proves Remark 10

Proof Because the neighbor radius 119903120578 is added to the repul-sion function as a variable the internal repulsion is

119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)

The neighbor distance 119889119894119895 can be considered a randomvariable with expectation E[119889119894119895] = 120588119894119895 thus we define 119875120588 asthe internal repulsion by expectation of neighbor distance asfollows119875120588 = 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)

Let 119890119875 be the tracking error of 119875120588 for the desired value 119875119889119890119875 = 119875119889 minus 119875120588 = 119875119889 minus 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)

Using the quadratic form of 119890119875 we define the Lyapunovcandidate function 119881as follows119881 = 12 (119875119889 minus 119875120588)2 = 121198901198752 (A33)

The derivative of 119881 is119889119881119889119905 = (119875119889 minus 119875120588) sdot (minus 119889119889119905 ( 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)

We set the derivative of 119903120578 to 119866119868119877119877119890119875 for a positive gain119866119868119877119877 and the differentiation of the repulsion function toalways be greater than zero for 0 lt 120588119894119895 lt 119903120578119889119903120578119889119905 = 119866119868119877119877119890119875120597119892 (119903120578 120588119894119895)120597119903120578 gt 0 (exist120588119894119895 120588119894119895 lt 119903120578) (A35)

22 Complexity

Therefore the derivative of 119881 is always smaller than zero forL = 0 119889119881119889119905 = minus1198661198681198771198771198902119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)

If the conditions of (A35) are satisfied 119881 converges tozero Thus the tracking error for internal repulsion by theexpectation of neighbor distance goes to zero119890119875 = 119875119889 minus 119875120588 997888rarr 0 as 119905 997888rarr infin (A37)

Data Availability

The simulation software and data files used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

The authors have no conflicts of interest to declare regardingthe publication of this paper

Acknowledgments

This research was financially supported by the Ministryof Trade Industry and Energy (MOTIE) and the KoreaEvaluation Institute of Industrial Technology (KEIT) throughthe Industrial Strategic Technology Development Program(10080636)

References

[1] J C Latombe Robot Motion Planning Kluwer Academic Pub-lishers 1991

[2] H Cheset K M Lynch S Hutchinson et al Principles ofRobot Motion eory Algorithms and Implementations TheMIT Press Cambridge UK 2005

[3] L E Kavaraki P Svestka J C Latmobe and M H OvermarsldquoProbabilistic roadmaps for path planning in high-dimensionalconfiguration spacerdquo IEEE Transactions on Robotics andAutomation vol 12 no 4 pp 566ndash580 1996

[4] J Barraquand L Kavraki J-C Latombe R Motwani T-Y Liand P Raghavan ldquoA random sampling scheme for path plan-ningrdquo International Journal of Robotics Research vol 16 no 6pp 759ndash774 1997

[5] J J Kuffner and S M LaValle ldquoRRT-Connect An EfficientApproach to Single-Query Path Planningrdquo in Proceedings of theIEEE ICRA vol 2 pp 995ndash1001 2000

[6] D Hsu J-C Latombe andH Kurniawati ldquoOn the probabilisticfoundations of probabilistic roadmap planningrdquo InternationalJournal of Robotics Research vol 25 no 7 pp 627ndash643 2006

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

[8] J D Marble and K E Bekris ldquoAsymptotically near-optimalplanning with probabilistic roadmap spannersrdquo IEEE Transac-tions on Robotics vol 29 no 2 pp 432ndash444 2013

[9] D Balkcom A Kannan Y-H Lyu W Wang and Y ZhangldquoMetric Cells Towards Complete Search for Optimal Tra-jectoriesrdquo in Proceedings of the 2015 IEEERSJ InternationalConference on Intelligent Robots and Systems IROS 2015 pp4941ndash4948 Germany October 2015

[10] M Elbanhawi and M Simic ldquoSampling-based robot motionplanning a reviewrdquo IEEE Access vol 2 pp 56ndash77 2014

[11] Z Sun D Hsu T Jiang H Kurniawati and J H Reif ldquoNarrowpassage sampling for probabilistic roadmap planningrdquo IEEETransactions on Robotics vol 21 no 6 pp 1105ndash1115 2005

[12] V Boor M H Overmars and A F van der Stappen ldquoTheGaussian sampling strategy for probabilistic roadmapplannersrdquoin Proceedings of the IEEE International Conference on Roboticsand Automation vol 2 pp 1018ndash1023 1999

[13] S A Wilmarth N M Amato and P F Stiller ldquoMAPRM Aprobabilistic roadmapplannerwith sampling on themedial axisof the spacerdquo inProceedings of the IEEE International Conferenceon Robotics and Automation vol 2 pp 1024ndash1031 1999

[14] T Simeon J-P Laumond and C Nissoux ldquoVisibility-basedprobabilistic roadmaps for motion planningrdquo AdvancedRobotics vol 14 no 6 pp 477ndash493 2000

[15] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEETransactions on Automatic Control vol 49 no 9 pp 1520ndash1533 2004

[16] J Yan and H Yu ldquoDistributed Optimization of Multiagent Sys-tems inDirectedNetworkswith Time-VaryingDelayrdquo Journal ofControl Science and Engineering vol 2017 Article ID 7937916 9pages 2017

[17] S Weerakkody X Liu S H Son and B Sinopoli ldquoA graph-theoretic characterization of perfect attackability for securedesign of distributed control systemsrdquo IEEE Transactions onControl of Network Systems vol 4 no 1 pp 60ndash70 2017

[18] R Zhang and Q Zhu ldquoSecure and resilient distributedmachinelearning under adversarial environmentsrdquo in Proceedings of the18th International Conference on Information Fusion pp 644ndash651 2015

[19] S Sundaram and B Gharesifard ldquoDistributed OptimizationUnder Adversarial Nodesrdquo IEEE Transactions on AutomaticControl 2018

[20] J Cortes and F Bullo ldquoCoordination and geometric opti-mization via distributed dynamical systemsrdquo SIAM Journal onControl and Optimization vol 44 no 5 pp 1543ndash1574 2005

[21] A Kwok and SMartinez ldquoUnicycle coverage control via hybridmodelingrdquo Institute of Electrical andElectronics EngineersTrans-actions on Automatic Control vol 55 no 2 pp 528ndash532 2010

[22] H Mahboubi and A G Aghdam ldquoDistributed deploymentalgorithms for coverage improvement in a network of wirelessmobile sensors relocation by virtual forcerdquo IEEE Transactionson Control of Network Systems vol 4 no 4 pp 736ndash748 2017

[23] J-H Park J-H Bae and M-H Baeg ldquoAdaptation algorithmof geometric graphs for robot motion planning in dynamicenvironmentsrdquo Mathematical Problems in Engineering vol2016 Article ID 3973467 19 pages 2016

[24] httpswwwyaskawaeucomenproductsroboticmotoman-robotsproductdetailproductmpx3500

[25] K Ferland Discrete Mathematics CENGAGE Learning Bel-mont NY USA 2008

[26] T Tao An Introduction to Measure eory American Mathe-matical Society 2011

Complexity 23

[27] D P Bertsekas Constrained Optimization and Lagrange Mul-tiplier Methods Athena Scientific Massachusetts MASS USA1996

[28] D P Bertsekas and J N Tsitsiklis Parallel andDistributed Com-putation Numerical Methods Athena Scientific MassachusettsMASS USA 2014

[29] C Langbort L Xiao R DrsquoAndrea and S Boyd ldquoA decomposi-tion approach to distributed analysis of networked systemsrdquo inProceedings of the 2004 43rd IEEE Conference on Decision andControl (CDC) pp 3980ndash3985 Bahamas CaribbeanDecember2004

[30] J N Tsitsiklis D P Bertsekas and M Athans ldquoDistributedasynchronous deterministic and stochastic gradient optimiza-tion algorithmsrdquo IEEE Transactions on Automatic Control vol31 no 9 pp 803ndash812 1986

[31] P Sadegh ldquoConstrained optimization via stochastic approxima-tion with a simultaneous perturbation gradient approximationrdquoAutomatica vol 33 no 5 pp 889ndash892 1997

[32] K Marti Stochastic Optimization Methods Springer BerlinHeidelberg 3rd edition 2008

[33] R H Myers D C Montgomery and C M Anderson-CookResponse Surface Methodology Process and Product in Opti-mization Using Designed Experiments John Wiley amp Sons 3rdedition 2009

[34] M S Ebeida S AMitchell A Patney A A Davidson and J DOwens ldquoA simple algorithm formaximal poisson-disk samplingin high dimensionsrdquo Computer Graphics Forum vol 31 no 2pp 785ndash794 2012

[35] E A Jackson Equilibrium Statistical Mechanics Prentice-HallNew Jersey NJ USA 1968

[36] L Kocarev Consensus and Synchronization in Complex Net-works Springer Berlin Heidelberg 2013

[37] A Barrat M Barthelemy and A Vespignani DynamicalProcesses on Complex Networks Cambridge University PressCambridge UK 2008

[38] C EricsonReal-TimeCollisionDetection CRCPress NewYorkNY USA 2004

[39] W F BasenerTopology and Its Applications JohnWiley amp SonsHoboken NJ USA 2006

[40] X Wang ldquoVolumes of Generalized Unit Ballsrdquo MathematicsMagazine vol 78 no 5 pp 390ndash395 2005

Hindawiwwwhindawicom Volume 2018

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Dierential EquationsInternational Journal of

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Decision SciencesAdvances in


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Submit your manuscripts atwwwhindawicom

2 Complexity

algorithms have been proposed to overcome drawbacks inthe uniform sampling method Nevertheless the effect ofsampling strategies on optimal motion planning remainsunclear The node sampling strategy that fully captures theshape of the free configuration space remains an importantproblem in optimal motion planning [9 10] In the queryphase of the roadmapmethod the shortest path is obtained asthe optimal path using graph informationThus constructingan appropriate graph is closely related to the problem ofoptimal motion planning

Another area highly relevant to our work here is researchon optimization techniques In recent years with grow-ing interest in machine learning and Artificial Intelligencetechniques of distributed optimization that can handlelarge amounts of data and high-dimensional variables haveemerged as an important issue In [15 16] distributed opti-mization techniques were considered formulti-agent systemsin complex networks Secure and resilient techniques [17ndash19] have also been developed to guarantee the stability ofdistributed algorithms in adversarial environments As highlyrelevant to ourwork here coveragemaximizationwas studiedusing methods of distributed optimization in applications ofsensor networks [20ndash22]

This paper proposes an algorithm to optimize theroadmap graph that can cover arbitrary morphologies ofthe free configuration space to maximize coverage Ourprevious study [23] considered an adaptation algorithm forgeometric graphs in an intuitive manner However in thisstudy we reanalyze that adaptive graph algorithm and presentits results in terms of the maximization of graph coverageWefound that the optimization algorithm updates the positionsof the nodes along the direction of expanding the region ofcovering until it reaches equilibrium where expansion andcontraction factors are balanced The results of a simulationto test the proposed optimization algorithm show that it canconstruct a suitable graph to describe arbitrary configurationspaces and the resulting graph can generate paths of shorterlength for various test queries

2 Problem Statement

The configuration space Q sub R119899 is the space of all possibleconfigurations of the system where n is the number ofdegrees of freedom (DOF) of the system and R119899 is n-dimensional Euclidean space [1] In the formulation of theconfiguration space the robot can be abstracted as a point q isinQ Free configuration is defined as one where the robot 119877(q)does not intersect with any obstacle in the workspaceThe setof all free configurations is defined as the free configurationspace Q119891119903119890119890 sub Q and it can be determined by checking forcollisions between obstacles and the robot in the workspaceEvery workspace obstacle 119882119874119894 is mapped as an obstacle inthe configuration space119876119874119894 = q isin Q | 119877(q) cap119882119874119894 = 120593 [2]Figure 1 shows the mapping relations between the workspaceand the configuration space for a two-link articulated robot

The problem of motion planning requires as solution afeasible path in the free configuration space for a specifiedinitial motion and a goal The path planning problem can thenbe defined as follows

Definition 1 ((path planning problem) [1]) Given a set ofcollision-free configurations Q119891119903119890119890 sub Q and initial and goalconfigurations q119894119899119894119905 q119892119900119886119897 isin Q119891119903119890119890 find a continuous curve120590 isin Σ = 119905 | 119905 [0 1] 997888rarr Q119891119903119890119890 where 120590(0) = q119894119899119894119905 and120590(1) = q119892119900119886119897

In this study we employ the roadmap method for pathplanning The roadmap represents the configuration spacetopologically by a network of one-dimensional (1D) curvesIt is defined as follows

Definition 2 ((roadmap) [2]) A union of 1D curves is aroadmap RM if for all q119894119899119894119905 and q119892119900119886119897 in Q119891119903119890119890 that can beconnected by a path the following properties hold

(1) Accessibility there exists a path from q119894119899119894119905 isin Q119891119903119890119890 tosome q1015840119894119899119894119905 isin RM

(2) Departability there exists a path from q1015840119892119900119886119897 isin RMto q119892119900119886119897 isin Q119891119903119890119890

(3) Connectivity there exists a path in RM between q1015840119894119899119894119905and q1015840119892119900119886119897

The result of the roadmap method is a graph G = (VE)consisting of a set of 119873 nodes V = q119894 isin Q119891119903119890119890 | forall119894 isin1 sdot sdot sdot 119873 and a set of edges E = 119890119894119895 = (q119894 q119895) | q119894 q119895 isin VEdge 119890119894119895 is created if the line segment between nodes q119894 andq119895 persists in Q119891119903119890119890 ie 119905q119894 + (1 minus 119905)q119895 isin Q119891119903119890119890 forall119905 isin [0 1]

To obtain a feasible path for the robotrsquos motion usinginformation from the roadmap graph the three propertiesin Definition 2 need to be met Furthermore for optimalplanning the best roadmap must be determined among a setof feasible roadmaps We now employ roadmap coverage asa performance index to assess the optimality of the roadmapgraph

To describe roadmap coverage the covering region of nodeq119894 for a certain neighbor radius 119903120578 is defined as follows

B119894 = B (q119894 119903120578) = q isin R119899 | 1003817100381710038171003817q minus q11989410038171003817100381710038172 le 1199031205782 q119894 isin V (1)

Let B = B1B2 sdot sdot sdot B119873 be the family of sets of B119894subsequently the roadmap coverage for neighbor radius 119903120578 isdefined as the union of B119894s as follows

Definition 3 (roadmap coverage) Roadmap coverage C forroadmap G is the region of the union of the family of setsB

C (G) = ⋃B = B1 ⋃B2 ⋃ sdot sdot sdot⋃B119873 (2)

Figure 2 shows an example of a roadmap and its coveragefor a configuration space

Let 120583(sdot) be the Lebesgue measure by the inclusionndashexclusion principle [25 26] the volume of roadmap coverage120583(C)can be represented as follows120583 (C (G)) = 120583 (⋃B) = 119873sum

119894=1(minus1)119894minus1 119878119894 (3)

where 119878119894 = sum1le1198951lt1198952sdotsdotsdotlt119895119894le119873 120583(B1198951 cap B1198952 cap sdot sdot sdot cap B119895119894)

Complexity 3

Collision motion

R(q)

Workspaceobstacle

Configuration space

Robotkinematics

Workspace Workspace

Workspace

Workspace

Workspace Workspace

QOi

free

WOi



free

ir

2


r

2 i isin |




4 Complexity







(minus1)119894minus1 119878119894= 1198781 + 119873sum

119894=2(minus1)119894minus1 119878119894 = 119873sum

119894=1120583 (B119894) minus 119868 (B) (5)







120583 (B119894 cap B119895)) (7)







sum1le119894lt119895le119873





sum1le119894lt119895le119873





Φ(q119894 q119895) = 2 sdot int1199031205782

1198891198941198952








119894=1 120582119894nabla119904(q119894) = 0



nablaΦ (q119894 [119896] q119895 [119896])+ 119873sum

119894=1120582119894 [119896] nabla119904 (q119894 [119896]) (13)

Complexity 5

Neighboring nodes

Neighbor radius

Node

r

i


















(20)

6 Complexity

minus66

minus4

4

minus2

6

0

2 4

2

0 2

4

0

6


(a)

minus88

minus6

minus4

6

minus2

84

0

62

2

4

4

0 2

6

minus2 0

8






q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894

119892 (119889119894119895 [119896]) (q119894 [119896] minus q119895 [119896])minus 120576120582119894 [119896]M+w119894 [119896] (22)






L = [119871 119894119895] 119871 119894119895 = minus119892 (119889119894119895) 119894 = 119895

119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (23)

Complexity 7

(a) (b)


















0] (28)


8 Complexity

2 1-1 1

rS








+


1001]]]]]]0]]]]]]]]= [ 04167minus04167]

(29)









119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 (31)



119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 = 12 119873sum

119894=12 119873sum119895=1119894 =119895

10038161003816100381610038161003816119892 (119889119894119895)10038161003816100381610038161003816= 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119889119894119895) (32)


Complexity 9

05833 08334 05833

-04167 04167

2 1





]

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a)

minus1

minus05

0

05

1

minus1minus05

005

1x1

x2

005

115

225

335

cost

(b)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2


(c)

05

1

15

2

25

3

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2





10 Complexity

1


002040608

1

minus1 04

02

06

08

minus02

minus08

minus04

minus06 0

(a)


002040608

1

x2

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1

x1(b)


002040608

1

x2

0 1minus1 02

06

08

04

minus02

minus06

minus08

minus04

x1(c)

minus1

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1


002040608

1

x1

x2

0020406081121416182

(d)


0

1000

2000

3000

4000

5000

6000

7000

8000

9000

57 59 61 63 65 67 69


minimummaximum


1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55

Neighbor radius

Inte

rnal

repu

lsion

Not converge



Complexity 11

(a) (b) (c) (d)








119894=1 sum119873119895=1119894 =119895 119892(119903120578E[119889119894119895])



(35)


119897=1198960120576 sdot (119875119889 minus 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119903120578 [119897] 120588119894119895 [119897]))+ 119903120578 [1198960] (36)




12 Complexity

0

1000

2000

3000

4000

5000

6000

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201





w[k]


Roadmap

+

minus



Error Controller


Edgeconstruction

x[k]

x[k + 1]+

minus V

E

G = (V E)

Node

Edge

Unit delay

P =1

21

zminus1

nN + [k] otimes n

r


(N otimes+)

PdeP

GIRR int eP dt





Complexity 13

Initialize nodes



Start

Resume threads

Synchronize

Terminate

Quit


Parallel Processing



Update node

Update edge

N

Y

Timer event


At each node



i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]


l=k0



i[k] = sum[k]isin




6 Case Study




14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3

Collision motion

R(q)

Workspaceobstacle

Configuration space

Robotkinematics

Workspace Workspace

Workspace

Workspace

Workspace Workspace

QOi

free

WOi



free

ir

2


r

2 i isin |




4 Complexity







(minus1)119894minus1 119878119894= 1198781 + 119873sum

119894=2(minus1)119894minus1 119878119894 = 119873sum

119894=1120583 (B119894) minus 119868 (B) (5)







120583 (B119894 cap B119895)) (7)







sum1le119894lt119895le119873





sum1le119894lt119895le119873





Φ(q119894 q119895) = 2 sdot int1199031205782

1198891198941198952








119894=1 120582119894nabla119904(q119894) = 0



nablaΦ (q119894 [119896] q119895 [119896])+ 119873sum

119894=1120582119894 [119896] nabla119904 (q119894 [119896]) (13)

Complexity 5

Neighboring nodes

Neighbor radius

Node

r

i


















(20)

6 Complexity

minus66

minus4

4

minus2

6

0

2 4

2

0 2

4

0

6


(a)

minus88

minus6

minus4

6

minus2

84

0

62

2

4

4

0 2

6

minus2 0

8






q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894

119892 (119889119894119895 [119896]) (q119894 [119896] minus q119895 [119896])minus 120576120582119894 [119896]M+w119894 [119896] (22)






L = [119871 119894119895] 119871 119894119895 = minus119892 (119889119894119895) 119894 = 119895

119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (23)

Complexity 7

(a) (b)


















0] (28)


8 Complexity

2 1-1 1

rS








+


1001]]]]]]0]]]]]]]]= [ 04167minus04167]

(29)









119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 (31)



119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 = 12 119873sum

119894=12 119873sum119895=1119894 =119895

10038161003816100381610038161003816119892 (119889119894119895)10038161003816100381610038161003816= 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119889119894119895) (32)


Complexity 9

05833 08334 05833

-04167 04167

2 1





]

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a)

minus1

minus05

0

05

1

minus1minus05

005

1x1

x2

005

115

225

335

cost

(b)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2


(c)

05

1

15

2

25

3

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2





10 Complexity

1


002040608

1

minus1 04

02

06

08

minus02

minus08

minus04

minus06 0

(a)


002040608

1

x2

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1

x1(b)


002040608

1

x2

0 1minus1 02

06

08

04

minus02

minus06

minus08

minus04

x1(c)

minus1

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1


002040608

1

x1

x2

0020406081121416182

(d)


0

1000

2000

3000

4000

5000

6000

7000

8000

9000

57 59 61 63 65 67 69


minimummaximum


1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55

Neighbor radius

Inte

rnal

repu

lsion

Not converge



Complexity 11

(a) (b) (c) (d)








119894=1 sum119873119895=1119894 =119895 119892(119903120578E[119889119894119895])



(35)


119897=1198960120576 sdot (119875119889 minus 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119903120578 [119897] 120588119894119895 [119897]))+ 119903120578 [1198960] (36)




12 Complexity

0

1000

2000

3000

4000

5000

6000

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201





w[k]


Roadmap

+

minus



Error Controller


Edgeconstruction

x[k]

x[k + 1]+

minus V

E

G = (V E)

Node

Edge

Unit delay

P =1

21

zminus1

nN + [k] otimes n

r


(N otimes+)

PdeP

GIRR int eP dt





Complexity 13

Initialize nodes



Start

Resume threads

Synchronize

Terminate

Quit


Parallel Processing



Update node

Update edge

N

Y

Timer event


At each node



i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]


l=k0



i[k] = sum[k]isin




6 Case Study




14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity







(minus1)119894minus1 119878119894= 1198781 + 119873sum

119894=2(minus1)119894minus1 119878119894 = 119873sum

119894=1120583 (B119894) minus 119868 (B) (5)







120583 (B119894 cap B119895)) (7)







sum1le119894lt119895le119873





sum1le119894lt119895le119873





Φ(q119894 q119895) = 2 sdot int1199031205782

1198891198941198952








119894=1 120582119894nabla119904(q119894) = 0



nablaΦ (q119894 [119896] q119895 [119896])+ 119873sum

119894=1120582119894 [119896] nabla119904 (q119894 [119896]) (13)

Complexity 5

Neighboring nodes

Neighbor radius

Node

r

i


















(20)

6 Complexity

minus66

minus4

4

minus2

6

0

2 4

2

0 2

4

0

6


(a)

minus88

minus6

minus4

6

minus2

84

0

62

2

4

4

0 2

6

minus2 0

8






q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894

119892 (119889119894119895 [119896]) (q119894 [119896] minus q119895 [119896])minus 120576120582119894 [119896]M+w119894 [119896] (22)






L = [119871 119894119895] 119871 119894119895 = minus119892 (119889119894119895) 119894 = 119895

119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (23)

Complexity 7

(a) (b)


















0] (28)


8 Complexity

2 1-1 1

rS








+


1001]]]]]]0]]]]]]]]= [ 04167minus04167]

(29)









119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 (31)



119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 = 12 119873sum

119894=12 119873sum119895=1119894 =119895

10038161003816100381610038161003816119892 (119889119894119895)10038161003816100381610038161003816= 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119889119894119895) (32)


Complexity 9

05833 08334 05833

-04167 04167

2 1





]

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a)

minus1

minus05

0

05

1

minus1minus05

005

1x1

x2

005

115

225

335

cost

(b)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2


(c)

05

1

15

2

25

3

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2





10 Complexity

1


002040608

1

minus1 04

02

06

08

minus02

minus08

minus04

minus06 0

(a)


002040608

1

x2

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1

x1(b)


002040608

1

x2

0 1minus1 02

06

08

04

minus02

minus06

minus08

minus04

x1(c)

minus1

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1


002040608

1

x1

x2

0020406081121416182

(d)


0

1000

2000

3000

4000

5000

6000

7000

8000

9000

57 59 61 63 65 67 69


minimummaximum


1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55

Neighbor radius

Inte

rnal

repu

lsion

Not converge



Complexity 11

(a) (b) (c) (d)








119894=1 sum119873119895=1119894 =119895 119892(119903120578E[119889119894119895])



(35)


119897=1198960120576 sdot (119875119889 minus 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119903120578 [119897] 120588119894119895 [119897]))+ 119903120578 [1198960] (36)




12 Complexity

0

1000

2000

3000

4000

5000

6000

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201





w[k]


Roadmap

+

minus



Error Controller


Edgeconstruction

x[k]

x[k + 1]+

minus V

E

G = (V E)

Node

Edge

Unit delay

P =1

21

zminus1

nN + [k] otimes n

r


(N otimes+)

PdeP

GIRR int eP dt





Complexity 13

Initialize nodes



Start

Resume threads

Synchronize

Terminate

Quit


Parallel Processing



Update node

Update edge

N

Y

Timer event


At each node



i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]


l=k0



i[k] = sum[k]isin




6 Case Study




14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5

Neighboring nodes

Neighbor radius

Node

r

i


















(20)

6 Complexity

minus66

minus4

4

minus2

6

0

2 4

2

0 2

4

0

6


(a)

minus88

minus6

minus4

6

minus2

84

0

62

2

4

4

0 2

6

minus2 0

8






q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894

119892 (119889119894119895 [119896]) (q119894 [119896] minus q119895 [119896])minus 120576120582119894 [119896]M+w119894 [119896] (22)






L = [119871 119894119895] 119871 119894119895 = minus119892 (119889119894119895) 119894 = 119895

119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (23)

Complexity 7

(a) (b)


















0] (28)


8 Complexity

2 1-1 1

rS








+


1001]]]]]]0]]]]]]]]= [ 04167minus04167]

(29)









119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 (31)



119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 = 12 119873sum

119894=12 119873sum119895=1119894 =119895

10038161003816100381610038161003816119892 (119889119894119895)10038161003816100381610038161003816= 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119889119894119895) (32)


Complexity 9

05833 08334 05833

-04167 04167

2 1





]

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a)

minus1

minus05

0

05

1

minus1minus05

005

1x1

x2

005

115

225

335

cost

(b)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2


(c)

05

1

15

2

25

3

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2





10 Complexity

1


002040608

1

minus1 04

02

06

08

minus02

minus08

minus04

minus06 0

(a)


002040608

1

x2

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1

x1(b)


002040608

1

x2

0 1minus1 02

06

08

04

minus02

minus06

minus08

minus04

x1(c)

minus1

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1


002040608

1

x1

x2

0020406081121416182

(d)


0

1000

2000

3000

4000

5000

6000

7000

8000

9000

57 59 61 63 65 67 69


minimummaximum


1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55

Neighbor radius

Inte

rnal

repu

lsion

Not converge



Complexity 11

(a) (b) (c) (d)








119894=1 sum119873119895=1119894 =119895 119892(119903120578E[119889119894119895])



(35)


119897=1198960120576 sdot (119875119889 minus 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119903120578 [119897] 120588119894119895 [119897]))+ 119903120578 [1198960] (36)




12 Complexity

0

1000

2000

3000

4000

5000

6000

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201





w[k]


Roadmap

+

minus



Error Controller


Edgeconstruction

x[k]

x[k + 1]+

minus V

E

G = (V E)

Node

Edge

Unit delay

P =1

21

zminus1

nN + [k] otimes n

r


(N otimes+)

PdeP

GIRR int eP dt





Complexity 13

Initialize nodes



Start

Resume threads

Synchronize

Terminate

Quit


Parallel Processing



Update node

Update edge

N

Y

Timer event


At each node



i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]


l=k0



i[k] = sum[k]isin




6 Case Study




14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity

minus66

minus4

4

minus2

6

0

2 4

2

0 2

4

0

6


(a)

minus88

minus6

minus4

6

minus2

84

0

62

2

4

4

0 2

6

minus2 0

8






q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894

119892 (119889119894119895 [119896]) (q119894 [119896] minus q119895 [119896])minus 120576120582119894 [119896]M+w119894 [119896] (22)






L = [119871 119894119895] 119871 119894119895 = minus119892 (119889119894119895) 119894 = 119895

119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (23)

Complexity 7

(a) (b)


















0] (28)


8 Complexity

2 1-1 1

rS








+


1001]]]]]]0]]]]]]]]= [ 04167minus04167]

(29)









119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 (31)



119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 = 12 119873sum

119894=12 119873sum119895=1119894 =119895

10038161003816100381610038161003816119892 (119889119894119895)10038161003816100381610038161003816= 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119889119894119895) (32)


Complexity 9

05833 08334 05833

-04167 04167

2 1





]

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a)

minus1

minus05

0

05

1

minus1minus05

005

1x1

x2

005

115

225

335

cost

(b)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2


(c)

05

1

15

2

25

3

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2





10 Complexity

1


002040608

1

minus1 04

02

06

08

minus02

minus08

minus04

minus06 0

(a)


002040608

1

x2

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1

x1(b)


002040608

1

x2

0 1minus1 02

06

08

04

minus02

minus06

minus08

minus04

x1(c)

minus1

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1


002040608

1

x1

x2

0020406081121416182

(d)


0

1000

2000

3000

4000

5000

6000

7000

8000

9000

57 59 61 63 65 67 69


minimummaximum


1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55

Neighbor radius

Inte

rnal

repu

lsion

Not converge



Complexity 11

(a) (b) (c) (d)








119894=1 sum119873119895=1119894 =119895 119892(119903120578E[119889119894119895])



(35)


119897=1198960120576 sdot (119875119889 minus 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119903120578 [119897] 120588119894119895 [119897]))+ 119903120578 [1198960] (36)




12 Complexity

0

1000

2000

3000

4000

5000

6000

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201





w[k]


Roadmap

+

minus



Error Controller


Edgeconstruction

x[k]

x[k + 1]+

minus V

E

G = (V E)

Node

Edge

Unit delay

P =1

21

zminus1

nN + [k] otimes n

r


(N otimes+)

PdeP

GIRR int eP dt





Complexity 13

Initialize nodes



Start

Resume threads

Synchronize

Terminate

Quit


Parallel Processing



Update node

Update edge

N

Y

Timer event


At each node



i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]


l=k0



i[k] = sum[k]isin




6 Case Study




14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7

(a) (b)


















0] (28)


8 Complexity

2 1-1 1

rS








+


1001]]]]]]0]]]]]]]]= [ 04167minus04167]

(29)









119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 (31)



119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 = 12 119873sum

119894=12 119873sum119895=1119894 =119895

10038161003816100381610038161003816119892 (119889119894119895)10038161003816100381610038161003816= 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119889119894119895) (32)


Complexity 9

05833 08334 05833

-04167 04167

2 1





]

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a)

minus1

minus05

0

05

1

minus1minus05

005

1x1

x2

005

115

225

335

cost

(b)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2


(c)

05

1

15

2

25

3

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2





10 Complexity

1


002040608

1

minus1 04

02

06

08

minus02

minus08

minus04

minus06 0

(a)


002040608

1

x2

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1

x1(b)


002040608

1

x2

0 1minus1 02

06

08

04

minus02

minus06

minus08

minus04

x1(c)

minus1

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1


002040608

1

x1

x2

0020406081121416182

(d)


0

1000

2000

3000

4000

5000

6000

7000

8000

9000

57 59 61 63 65 67 69


minimummaximum


1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55

Neighbor radius

Inte

rnal

repu

lsion

Not converge



Complexity 11

(a) (b) (c) (d)








119894=1 sum119873119895=1119894 =119895 119892(119903120578E[119889119894119895])



(35)


119897=1198960120576 sdot (119875119889 minus 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119903120578 [119897] 120588119894119895 [119897]))+ 119903120578 [1198960] (36)




12 Complexity

0

1000

2000

3000

4000

5000

6000

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201





w[k]


Roadmap

+

minus



Error Controller


Edgeconstruction

x[k]

x[k + 1]+

minus V

E

G = (V E)

Node

Edge

Unit delay

P =1

21

zminus1

nN + [k] otimes n

r


(N otimes+)

PdeP

GIRR int eP dt





Complexity 13

Initialize nodes



Start

Resume threads

Synchronize

Terminate

Quit


Parallel Processing



Update node

Update edge

N

Y

Timer event


At each node



i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]


l=k0



i[k] = sum[k]isin




6 Case Study




14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity

2 1-1 1

rS








+


1001]]]]]]0]]]]]]]]= [ 04167minus04167]

(29)









119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 (31)



119894=1

119873sum119895=1

10038161003816100381610038161003816119871 11989411989510038161003816100381610038161003816 = 12 119873sum

119894=12 119873sum119895=1119894 =119895

10038161003816100381610038161003816119892 (119889119894119895)10038161003816100381610038161003816= 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119889119894119895) (32)


Complexity 9

05833 08334 05833

-04167 04167

2 1





]

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a)

minus1

minus05

0

05

1

minus1minus05

005

1x1

x2

005

115

225

335

cost

(b)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2


(c)

05

1

15

2

25

3

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2





10 Complexity

1


002040608

1

minus1 04

02

06

08

minus02

minus08

minus04

minus06 0

(a)


002040608

1

x2

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1

x1(b)


002040608

1

x2

0 1minus1 02

06

08

04

minus02

minus06

minus08

minus04

x1(c)

minus1

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1


002040608

1

x1

x2

0020406081121416182

(d)


0

1000

2000

3000

4000

5000

6000

7000

8000

9000

57 59 61 63 65 67 69


minimummaximum


1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55

Neighbor radius

Inte

rnal

repu

lsion

Not converge



Complexity 11

(a) (b) (c) (d)








119894=1 sum119873119895=1119894 =119895 119892(119903120578E[119889119894119895])



(35)


119897=1198960120576 sdot (119875119889 minus 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119903120578 [119897] 120588119894119895 [119897]))+ 119903120578 [1198960] (36)




12 Complexity

0

1000

2000

3000

4000

5000

6000

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201





w[k]


Roadmap

+

minus



Error Controller


Edgeconstruction

x[k]

x[k + 1]+

minus V

E

G = (V E)

Node

Edge

Unit delay

P =1

21

zminus1

nN + [k] otimes n

r


(N otimes+)

PdeP

GIRR int eP dt





Complexity 13

Initialize nodes



Start

Resume threads

Synchronize

Terminate

Quit


Parallel Processing



Update node

Update edge

N

Y

Timer event


At each node



i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]


l=k0



i[k] = sum[k]isin




6 Case Study




14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9

05833 08334 05833

-04167 04167

2 1





]

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

(a)

minus1

minus05

0

05

1

minus1minus05

005

1x1

x2

005

115

225

335

cost

(b)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2


(c)

05

1

15

2

25

3

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

x2





10 Complexity

1


002040608

1

minus1 04

02

06

08

minus02

minus08

minus04

minus06 0

(a)


002040608

1

x2

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1

x1(b)


002040608

1

x2

0 1minus1 02

06

08

04

minus02

minus06

minus08

minus04

x1(c)

minus1

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1


002040608

1

x1

x2

0020406081121416182

(d)


0

1000

2000

3000

4000

5000

6000

7000

8000

9000

57 59 61 63 65 67 69


minimummaximum


1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55

Neighbor radius

Inte

rnal

repu

lsion

Not converge



Complexity 11

(a) (b) (c) (d)








119894=1 sum119873119895=1119894 =119895 119892(119903120578E[119889119894119895])



(35)


119897=1198960120576 sdot (119875119889 minus 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119903120578 [119897] 120588119894119895 [119897]))+ 119903120578 [1198960] (36)




12 Complexity

0

1000

2000

3000

4000

5000

6000

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201





w[k]


Roadmap

+

minus



Error Controller


Edgeconstruction

x[k]

x[k + 1]+

minus V

E

G = (V E)

Node

Edge

Unit delay

P =1

21

zminus1

nN + [k] otimes n

r


(N otimes+)

PdeP

GIRR int eP dt





Complexity 13

Initialize nodes



Start

Resume threads

Synchronize

Terminate

Quit


Parallel Processing



Update node

Update edge

N

Y

Timer event


At each node



i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]


l=k0



i[k] = sum[k]isin




6 Case Study




14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity

1


002040608

1

minus1 04

02

06

08

minus02

minus08

minus04

minus06 0

(a)


002040608

1

x2

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1

x1(b)


002040608

1

x2

0 1minus1 02

06

08

04

minus02

minus06

minus08

minus04

x1(c)

minus1

minus08

minus06

minus04

minus02 0

02

04

06

08 1minus1


002040608

1

x1

x2

0020406081121416182

(d)


0

1000

2000

3000

4000

5000

6000

7000

8000

9000

57 59 61 63 65 67 69


minimummaximum


1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55

Neighbor radius

Inte

rnal

repu

lsion

Not converge



Complexity 11

(a) (b) (c) (d)








119894=1 sum119873119895=1119894 =119895 119892(119903120578E[119889119894119895])



(35)


119897=1198960120576 sdot (119875119889 minus 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119903120578 [119897] 120588119894119895 [119897]))+ 119903120578 [1198960] (36)




12 Complexity

0

1000

2000

3000

4000

5000

6000

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201





w[k]


Roadmap

+

minus



Error Controller


Edgeconstruction

x[k]

x[k + 1]+

minus V

E

G = (V E)

Node

Edge

Unit delay

P =1

21

zminus1

nN + [k] otimes n

r


(N otimes+)

PdeP

GIRR int eP dt





Complexity 13

Initialize nodes



Start

Resume threads

Synchronize

Terminate

Quit


Parallel Processing



Update node

Update edge

N

Y

Timer event


At each node



i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]


l=k0



i[k] = sum[k]isin




6 Case Study




14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11

(a) (b) (c) (d)








119894=1 sum119873119895=1119894 =119895 119892(119903120578E[119889119894119895])



(35)


119897=1198960120576 sdot (119875119889 minus 119873sum

119894=1

119873sum119895=1119894 =119895

119892 (119903120578 [119897] 120588119894119895 [119897]))+ 119903120578 [1198960] (36)




12 Complexity

0

1000

2000

3000

4000

5000

6000

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201





w[k]


Roadmap

+

minus



Error Controller


Edgeconstruction

x[k]

x[k + 1]+

minus V

E

G = (V E)

Node

Edge

Unit delay

P =1

21

zminus1

nN + [k] otimes n

r


(N otimes+)

PdeP

GIRR int eP dt





Complexity 13

Initialize nodes



Start

Resume threads

Synchronize

Terminate

Quit


Parallel Processing



Update node

Update edge

N

Y

Timer event


At each node



i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]


l=k0



i[k] = sum[k]isin




6 Case Study




14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity

0

1000

2000

3000

4000

5000

6000

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201





w[k]


Roadmap

+

minus



Error Controller


Edgeconstruction

x[k]

x[k + 1]+

minus V

E

G = (V E)

Node

Edge

Unit delay

P =1

21

zminus1

nN + [k] otimes n

r


(N otimes+)

PdeP

GIRR int eP dt





Complexity 13

Initialize nodes



Start

Resume threads

Synchronize

Terminate

Quit


Parallel Processing



Update node

Update edge

N

Y

Timer event


At each node



i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]


l=k0



i[k] = sum[k]isin




6 Case Study




14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 13

Initialize nodes



Start

Resume threads

Synchronize

Terminate

Quit


Parallel Processing



Update node

Update edge

N

Y

Timer event


At each node



i[k] = +i[k]

i[k + 1] = i[k] + i[k] minus i[k]


l=k0



i[k] = sum[k]isin




6 Case Study




14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








14 Complexity

S

L

U

T

B

R

(a) (b)








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 15

(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)



7 Conclusion


16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








16 Complexity

(a) (b)

RRT

PRMlowast

Proposed

(c) (d)



Appendix

A


1le119894lt119895le119873120583 (B119894 cap B119895)) (A1)








(minus1)119894 119878119894 le 119873sum119894=2

119878119894 (A3)

Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 17

(a)

(b)

(c)




(119873119895 )) sdot 1198782 (2 le 119894 le 119873) (A4)


119873sum119894=2

119878119894 le (1198732)minus1( 2∐119895=2

(119873119895 )) sdot 1198782 + (1198732)minus1

sdot ( 3∐119895=2


(119873119895 ))

sdot 1198782 = (1198732)minus1

sdot 2∐119895=2

(119873119895) + 3∐119895=2

(119873119895) + sdot sdot sdot + 119873∐119895=2


( 119894∐119895=2

(119873119895 )) sdot ( sum1le119894lt119895le119873

120583 (B119894 cap B119895)) (A5)

18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








18 Complexity

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(a)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(b)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(c)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(d)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(e)

0100200300400500600700800

1 2 3 4 5 6 7 8 9 10


RRT(f)


0

1000

2000

3000

4000

5000

6000

7000


50 100 150 200 250 300Number of nodes N


Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 19





= (119873119894 )sum119896=1

120583 (C119894 (119896)) (A7)


for 0 le 120572119894+1 (119896) le 1 (A8)


119878119894+1 = ( 119873119894+1 )sum119896=1

120583 (C119894+1 (119896)) = ( 119873119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119897119896)) (A9)


119894+1 )sum119896=1

120572119894+1 (119896) sdot 120583 (C119894 (119896))= (119873119894 )sum

119897=1120573119894+1 (119897) sdot 120583 (C119894 (119897)) (A10)

wheresum( 119873119894+1 )



as follows

119878119894+1 = (119873119894 )sum119897=1

120573119894+1 (119897) sdot 120583 (C119894 (119897)) = 120573119879119894+1 sdot 120583119894 (A11)


1

2Φ(i j)

j

r

2

2

i i minus j = 2


1003817100381710038171003817120573119894+110038171003817100381710038171 = (119873119894 )sum119896=1

120573119894+1 (119896) = ( 119873119894+1 )sum119896=1




Γ (1 + (119899 minus 1) 2)119903119899minus1 (A14)



= int1199031205782

1198891198941198952



20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








20 Complexity



Φ(q119894q119895) = 2 sdot int1199031205782

1198891198941198952







[[[[[[[[119904 (q119894 + Δq1)119904 (q119894 + Δq2)119904 (q119894 + Δq119899119904)

]]]]]]]]= [[[[[[[[


= [[[[[[[[1 Δq11987911 Δq1198792 1 Δq119879119899119904

]]]]]]]][1205930120593]

(A19)




119868 = (M119879119868M119868)minus1M119879

119868w119894 (A20)


The matrix M119879119868M119868 is

M119879119868M119868 = [1 sdot sdot sdot 1

M119879 ][[[[[1 M1 ]]]]]

= [[[[[119899119878sum119898=1

1 119899119878sum119898=1

Δq119879119898119899119878sum119898=1

Δq119898 M119879M

]]]]] (A21)

if sum119899119904119898=1 Δq119898 = 0 then M119879

119868M119868 = [ 119899119878 0119879


becomes

119868 = [1205930] = [119899119878 0119879


M119879 ]w119894

= [[[119899minus1119878 sdot 119899119878sum119898=1

119904 (q119894 + Δq119898)(M119879M)minus1M119879w119894

]]] (A22)



q119894 [119896 + 1] = q119894 [119896]+ 120576 sumq119895[119896]isin120578119894



Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 21


q119894 [119896 + 1] = q119894 [119896]+ 120576 119873sum119895=1119894 =119895


119892 (119889119894119895 [119896]))q119894 [119896]minus 119873sum

119895=1119894 =119895119892 (119889119894119895 [119896]) q119895 [119896]) minus 120576120582M+w119894 [119896]

(A25)



119873sum119896=1119894 =119896

119892 (119889119894119896) 119894 = 119895 (A26)





[[[[[[[q1 [119896 + 1]q2 [119896 + 1]q119873 [119896 + 1]

]]]]]]] = [[[[[[[q1 [119896]q2 [119896]q119873 [119896]

]]]]]]]+ 120576([[[[[[[

L1 [119896]L2 [119896]L119873 [119896]


w1 [119896]w2 [119896]w119873 [119896]

]]]]]]]

(A28)






119875 = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 119889119894119895) (A30)


119894=1

119873sum119895=1119894 =119895

119892 (119903120578E [119889119894119895]) = 119873sum119894=1

119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A31)


119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895) (A32)



119873sum119895=1119894 =119895

119892 (119903120578 120588119894119895)))= minus119890119875 sdot ( 119873sum

119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) sdot 119889119903120578119889119905 (A34)


22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








22 Complexity


119894=1

119873sum119895=1119894 =119895

120597119892120597119903120578) lt 0 (A36)


Data Availability




Acknowledgments


References



























Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 23






















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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