Liveness-Based RRT Algorithm for Autonomous …downloads.hindawi.com/journals/jat/2017/7816263.pdfpath planning algorithm, which is shown in Algorithm 1. It has the ability of quickly

Research ArticleLiveness-Based RRT Algorithm for Autonomous UnderwaterVehicles Motion Planning

Yang Li1 Fubin Zhang1 Demin Xu1 and Jiguo Dai2

1The School of Marine Science and Technology Northwestern Polytechnical University Xirsquoan 710072 China2Department of Mechanical Engineering Texas Tech University Lubbock TX 79409 USA

Correspondence should be addressed to Fubin Zhang zhangfbnwpueducn

Received 26 June 2017 Accepted 13 September 2017 Published 19 October 2017

Academic Editor Cheng S Chin

Copyright copy 2017 Yang Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Motion planning is a crucial basic issue in robotics which aims at driving vehicles or robots towards to a given destination withvarious constraints such as obstacles and limited resource This paper presents a new version of rapidly exploring random trees(RRT) that is liveness-based RRT (Li-RRT) to address autonomous underwater vehicles (AUVs) motion problem Differentfrom typical RRT we define an index of each node in the random searching tree called ldquolivenessrdquo in this paper to describe thepotential effectiveness during the expanding process We show that Li-RRT is provably probabilistic completeness as original RRTIn addition the expected time of returning a valid path with Li-RRT is obviously reduced To verify the efficiency of our algorithmnumerical experiments are carried out in this paper

1 Introduction

In the last decade motion planning for single robot ormultiple robot system has gained a lot of research interestas one of the primary problems in robot field In mostapplication cases it mainly involves the structured mobilityto drive the robots to the final destination given any initialstates [1] In general the motion planning algorithms can begenerally categorized into two types [2] The first ones returnthe positions along the path from start to the goal which arecalled path planning algorithms In addition the second onesthat is trajectory planning algorithms (also called motionplanning algorithms) return a sequence set of input spacethat drives the robot towards the destination

In recent years various algorithms have been proposedfor both path and trajectory planning problems Artificialpotential fields (APF) can be viewed as a powerfulmethod forpath planning which was firstly proposed in [3] AlthoughAPF suffers from the presence of local minimums it isadaptable in such cases with online intractable changes ofenvironment Meantime various of methods are proposedto avoid sticking into local minimums such as potentialfunctions without local minimums [4] and random searchmethods [5ndash7]

Another remarkable probabilistic method was proposedas Probabilistic Roadmap (PRM) [8ndash11]The key idea of PRMis to generate a connective random graph that comprisesvalid nodes selected randomly in the free configurationspace (C-space) Meanwhile RRT was presented as a singlequery incremental algorithm [12] PRM and RRT are shownwith more effectiveness especially facing high-dimensionalproblems Besides they are insensitive to the complexity ofthe problems RRT has provably probabilistic completenesswhichmeans that the probability that RRT cannot find a validpath from the initial to the goal position converges to zeroas the number of samples tends to infinity Also the conceptof probabilistic completeness is officially defined which is acrucial property although it is weaker than ldquocompletenessrdquo inthe field of sample based algorithms Consequently RRT hasbeen applied in many applications ranging from industrialproduction to military battle systems

Several RRT based algorithms are designed in orderto deal with different constraints or to provide solutionsmore effectively For example in [13ndash16] additional dynamicdifferential constraints are considered when extending therandom tree A kind of new variation space called tra-jectory parameter-space (TP-space) was presented in [17]

HindawiJournal of Advanced TransportationVolume 2017 Article ID 7816263 10 pageshttpsdoiorg10115520177816263

2 Journal of Advanced Transportation

Particularly each edge is kinematically feasible once it isgenerated by two points in TP-space In [18] the expandingprocess utilizes implicit flood-fill-like mechanism to avoidthe local minimumThus the expected time to find the solu-tion is reduced Because of the insensitivity to the dimensionof problems it is able to be used to search time invariantparameters [19] It is a fact that AUVs have been widelyapplied in the marine resources exploration and utilization[20ndash23] In order to map a scalar field mutual informationbased multidimensional RRT algorithm was presented formultiple AUVs in [15]

However RRT is proven to converge almost to a nonop-timal path [24] in which RRTlowast was detailed to be asymptoti-cally optimal That means RRTlowast will ldquoalmost surelyrdquo find theoptimal path as the number of samples tends to infinity RRTlowastapplies the reconstruction of the searching tree which alwayscontains the optimal subtree at each iteration Since then sev-eral RRTlowast based optimal planning algorithms were proposed[25ndash28] and applied in many fields [29 30] LBT-RRT wasproposed in [25] which is a combination algorithmwith RRTand RRTlowast It defines a variable 120598 to describe the ldquodistancerdquoaway from the optimal path In other words 120598 can be viewedas the comparison between RRT and RRTlowast If 120598 equals zeroit becomes RRTlowast essentially In [26] RRTlowast is refined tobe capable of replanning with dynamic environments In[27] the limitation that is requirements for an asymptoticaloptimal steer function during the treersquos expansion is provablyaddressed based on sparse data structure

From [31] we know that the sampling strategies havemuch effect on the efficiency of sampling-based algorithmsIn early researches typical choice is sampling uniformly inC-space [8 12 24] Every node in C-space is sampled withexactly the same probability by which the prior informationactually is not exploited For instance a new node intuitivelyprefers to be sampled away from the region where a largenumber of samples already exist And it certainly reduces theprobability that samples are generated in the case of narrowpassages [31] For this reason various heuristic methodolo-gies are designed Goal bias is generally utilized to enhanceextending towards the goal due to its effectiveness Medialaxis is combined with path planning algorithms to searchthe C-space with much more efficiency [8] On the contraryGaussian sampling strategy prefers the region nearby obsta-cles [10 32] In this view geometric shapes of obstacles areable to be quickly constructed Some other hybrid samplingstrategies are proposed to combine two or more methodsThese methods uniformly consider the information of theenvironment to refine the sampling probability howeverthe information of nodes is not totally utilized Specificallyregions where a number of nodes are already sampled canbe mostly considered with less worth In other words it ispreferred to sample in ldquonewrdquo regions

In this paper we demonstrate the expanding mechanismof the random tree and make an estimation of the expandingability of each node We present several new indicators inthe expanding procedure which are utilized in our new algo-rithm Also we detail the indicators quantitative descriptionsof the nodes Meanwhile we provide theoretical analysisand property of our algorithm with mathematical proof (ie

probabilistic completeness and outperforming compared toRRT) The proof is based on a technical notation attractionsequence and probability theory

The reminder of the paper is organized as followsSection 2 introduces the definition and notations in thispaper The indicators which reflect the growing of the treeand expanding ability are analyzed in Section 3 Then ournew algorithm that is Li-RRT is described in detail InSection 4 the analysis of Li-RRT is provided Section 5presents an application example for a planar mobile robot Infinal concluding remarks are drawn in Section 6

2 Preliminaries and Problem Formation

21 Problem Formulation Path planning problem can begenerally viewed as a search problem in the C-space 119862 sube R119899For instance the C-space of a planar two-wheel mobile robotcan be defined as 119862 sube R3 that is position (119909 119910) headingangle 120579 In this way any 119909 isin 119862 sube R3 can describe thestate of the mobile robot Additionally we define 119862obs sub 119862and 119862free = 119862 119862obs as the obstacle C-space and free C-space separately The path planning problems require a finitesequence of states in 119862free from an initial state to a given goalstate Here we present the basic path planning problem asdefined in [24]

Problem 1 (path planning problem [24]) The bounded C-space 119862 obstacle space 119862obs initial state 119909start isin 119862free goalregion 119883goal isin 119862free denoted as tuple 119909start 119862free 119883goalreturn a path 120585 [0 1] rarr 119862free such that 120585(0) = 119909start120585(1) isin 119883goal If there exist no feasible paths return failure

Facing path planning problems each element of dimen-sion is considered independently with no other constraintsHowever it is unrealistic inmost applications Specifically thekinodynamic of AUVs is restricted in case of uncontrollabil-ity It means that there exists corresponding available inputsequence 119906(119904) [0 1] rarr 119880 that drives the AUV from 119909start to119883goal Subsequently the path can also be viewed as a functionof input sequence 120585(119906(sdot)) 119906(sdot) rarr 119862free Formally we give thedefinition of trajectory planning problem for AUVs

Problem 2 (trajectory planning problem for AUVs) Given119909start 119862free 119883goal and kinodynamic constraint of AUVs =119891(119909 119906) return an available control sequence 119906 [0 1] rarr 119880such that the path 120585(119906(sdot)) = 119909 isin 119862free | 120585(119906(0)) = 119909init120585(119906(1)) isin 119862free 120585(119906(119904)) = int1

0[119909init + 119891(119909 119906)]119889119904

In this paper we focus on the trajectory planning problemfor AUVs Since the initial force has major impact on themotion of AUVs compared to ground vehicles the kinody-namic constraints are more complex

3 Algorithms

RRT was firstly introduced by Lavalle as a sampling-basedpath planning algorithm which is shown in Algorithm 1It has the ability of quickly searching or exploring the C-space by a random tree At each iteration the tree expands

Journal of Advanced Transportation 3

Search tree

Grid points

Node

Grid space

ＧＣＨ(dj )

Figure 1 Grid point distance from the nearest node in the tree 120575 is an adjustable grid spacing and the dashed arrow indicates the distance119889119895 from the grid points to their closet nodes in tree

Require C-space 119862 Free C-space 119862free Initial state119909start Goal region 119883goal max searching iteration 119870Ensure Feasible path 120591(1) 119881 larr(119909start) 119864 larr 0G larr (119881 119864) Initialization(2) for 119896 = 1 2 119870 do(3) 119909rand larr Sample(119896) Randomized sample in 119862free(4) G(119881 119864) larr Extend(G 119909rand) Extend random tree(5) end for(6) 120591 larr Export(G) Return the feasible path(7) return 120591

Algorithm 1 Body of basic RRT algorithm [33]

sequentially towards the goalWe refer readers tomore detailsabout RRT in [12]

The main branches of RRT are firstly constructed as itrapidly reaches the far corners of the square As samplesare incrementally added in the searching tree C-spaces aremostly covered by smaller branches gradually In otherwordsgiven any interval 120598 gt 0 C-space will be filled with nodesas the number of samples tends to infinity It means theprobability that there exists at least one node of the randomtree located in the goal set equals one as the number ofsamples tends to infinity if there exists one available path in119862free that is lim119899rarrinfinPforall119909 isin C exist1199091015840 isin 119909RRT |119909 minus 1199091015840| lt 120598 120598 gt0 = 131 Coverage of a Graph A kind of coverage measure of thetree in C-space was proposed in [34] We uniformly overlaygrids of 119899119892 points and spacing 120575 on 119862 Define the minimumdistance from each grid point 119895 to the nearest node in the treeas 119889119895 Thus min(119889119895 120575) can describe the radius of the largestball centered at the grid point which contains no nodes of thetree and adjacent grid points as shown in Figure 1 Given agraphG we define its coverage measure as in [34]

119888 (G) = 1120575119899119892sum119895=1

min (119889119895 120575)119899119892 (1)

where 1120575 removes the impact of different spacing distanceThe coverage of a tree can be viewed as the average of all nodesdistances normalized by the grid space 120575

The key idea of this measure is to describe the dispersionof all nodes which is a conception proposed from the MonteCarlomethod that indicates the unevenness of sample pointsPrimarily the cover performance is significantly better withsmaller coverage measure In other words expanding morewidely and uniformly may lead to smaller coverage measureIn an extreme case that is 119888(G) = 0 which is impracticableif there exists at least one node of the tree in every grid pointthe tree can be viewed well distributed in 119862free with givensolution of the grids Intuitively we may try to optimize 119888(G)with an acceptable computing complexity which is addressedin Section 34

32 Growth Speed of a Tree The growth of a tree is naturallydefined as the derivative of the coverage measure We denotethe number of nodes in the tree as 119899119905 and then define thegrowth of the tree as [34]

119892 (G) = minusΔ119888 (G)Δ119899119905 (2)

Because the differential form 119889119888(G)119889119899119905 is intractably formedand computed we utilize the difference form to indicate thegrowing speed of the tree Obviously 119892(G) is able to beused to test the nodes of tree appropriately Let 119892 denote thethreshold if 119892(G) drops below 119892 with new node V119894 it meansthat V119894 is almost worthless

33 Liveliness of Nodes The factors which illustrate expand-ing ability of nodes in a tree can be roughly categorized intotwo parts the external circumstance factors and the internalcircumstance factors

331 External Circumstance of Nodes External environmenthas significant influence on the expanding ability of eachnode If some node locates in a blind alley it has absolutely nocontribution on searching As shown in Figure 2 it describesthe scope of a parent node that could be extended to within


xk

u1

u2

u3

un

x1k+1

x2k+1

x3k+1

xnk+1

middot middot middot

Figure 2 Latent expansion with obstacles

time interval Δ119905 Without loss of generality we assume thatan appropriate discrete control set 119880 is defined and thecardinality of119880 is119898 that is119880 = 1199061 1199062 119906119898 Let 119909119896 denotethe parent and let 119909119906119894

119896+1 119894 = 1 119898 denote the successor of119909119896The superscript 119906119894means the control vector applied on 119909119896

where the gray areas represent the obstacles If 119909119906119894119896+1

locates in119862obs for all 119896 = 1 2 119898 it certainly means that 119909119896 is deadfor the concurrent tree Also this expansion by 119909119896 shouldbe considered totally useless By reducing useless nodes thesearching process can be obtained more efficiently

Based on that the available potential successor set of119909119896 is achieved by applying 119880 that is 119883119880119896+1 = 119909119906119894119896+1

|Collision Free(119909119896 119909119906119894119896+1) = TRUE 119894 = 1 2 119898 Mean-time we denote the corresponding available control vector as = 119906119894 | Collision Free(119909119896 119909119906119894119896+1) = TRUE 119894 = 1 2 119898We denote the cardinality of as 119898119886 and define collisiondetective index as Se(119909119896) = 119898119886119898 Note that if Se(119909119896) = 0 thepoint 119909119896 is a dead point If so 119909119896 is necessarily removed fromthe tree Meanwhile we could define the collision detectiveindex of a tree as

Se (G) = sum119899119897119896=1

Se (119909119896)119899119897 (3)

where 119899119897 denotes the number of nodes in a tree which isutilized to normalize the feature of the nodes The collisiondetective index of a tree is the average value of all componentnodes

It is a fact that the prior information about the environ-ment is unavailable in many applications In such cases theperformance of searching C-space is more important Se(G)defined in (3) may give a reasonable solution on the processof expansion of searching tree Generally we prefer the nodesthat make Se(G) as large as possible or larger than a threshold119878119890 because itmeans that the probability of reaching the targetwill stay high

Unfortunately it is commonly difficult to tell whether anode is stuck in a blind alley or just in a narrow passage Notethat it is still necessary to expand 119909119896 if119898119886 is not equal to zeroIn other words only 119909119896 | 119898119886 = 0 could be removed in caseof unexpected failure In this way the set of state 119909119896 | 119898119886 =0 can be viewed as the inevitable obstacles Unlike that thedead nodes in this paper are naturally cheap to compute

332 Internal Structure of Trees Internal structure of atree also influences the expanding ability of nodes Due tothe stochastic sampling strategy it may lead to crowd insome regions by random expanding In Figure 3(a) the graycircles with dotted lines are considered as regression nodeswhich are avoided by RRT-blossommethod proposed in [18]Although it may be necessary to explore the regression statessince the complex environment the other unexplored regionshave reasonable high priority

In Figure 3(a) the hollow white dots are possible newlyadded nodes of an identical parent The hollow white dotsembraced in the ellipse together with a solid black dot arecloser to the neighbor nodes than their parents Figure 3(b)shows all expansions in the tree which avoid being closer tothe existing structure This strategy ensures that the tree willhave a strong tendency to explore the whole C-space withinflated branches

We define the degree of a node as the number of itssuccessors which is denoted by 119889(119909119896) All offspring of a nodeis defined as

119889119904 (119909119896) = sum119909119894isinUs(119909119896)

119889 (119909119894) (4)

where the set Us(119909119896) is the union of all offspring of the node119909119896 It is obviously that a node with more successors andoffspring has less expanding ability Figure 4 illustrates a treersquosdistribution of degrees and offspring

The neighbors of a node are viewed as obstacles whichare shown in Figure 5 The neighbor nodes are viewedas circular or spherical obstacles with a fixed radiusDefine the number of neighbor collisions as 119898nei

119886 (119909119896) =sum119898119894=1 1Collision Free(119909119896 119909119906119894119896+1 = TRUE) where 1(sdot) equals1 if equation (sdot) holds

Considering all these factors we design a hybrid indicatorto illustrate the quantity of liveliness of each node Synthesiz-ing two external and internal factors the liveliness index ofeach node in a tree is defined in the form of

Li (119909119896)= Se (119909119896) exp (minus119898ℎ) exp [minus119889 (119909119896)] exp[minus119889119904 (119909119896)119899119905 ] (5)

where 119899119905 is the number of total nodes at current iterationIndeed (5) reflects the expanding ability of a specific nodeWe translate Li(119909119896) in logs denoted as LnLi

LnLi (119909119896) = ln Li (119909119896)= ln Se (119909119896) minus [119898ℎ + 119889 (119909119896) + 119889119904119896119899119905 ] (6)

Since log function has the same monotonicity LnLi(119909119896) candescribe the liveness of each node Particularly if 119909119896 is a deadpoint that is Se(119909119896) = 0 we set LnLi(119909119896) = minusinf

Obviously the liveness LnLi defined in (6) will beattached to each node once added to the random tree Thevalue of LnLi is calculated and updated when the expandingprocess will be called It can be viewed as a guidance thatdescribes a better expanding direction for the searching tree


(a) Regression expansions (b) Nonregressing expansions

Figure 3 RRT-blossom regression New point tends to stay away from the existing structure like the hollow circles

d(x3) = 0

ds(x3) = 0

d(x6) = 0

ds(x6) = 0

d(x2) = 3

ds(x2) = 3

3

6d(x7) = 0

ds(x7) = 0

7

d(x1) = 2

ds(x1) = 6

d(x4) = 1

ds(x4) = 1

d(x5) = 0

ds(x5) = 0

1

4

5

2

Figure 4 The degree of nodes in a tree

Obstacle Obstacle

Obstacle

xk

u1u2

un

x1kx2

k

xnk

Figure 5 Neighbor obstacles The radius of neighbor obstacle isgenerally obtained from experience

34 Li-RRT In this section we present the hybrid RRTalgorithm that is Li-RRT as shown in Algorithm 2 It utilizesthe liveness of each node to guide the expanding process ofthe random searching treeMore efficient or useful nodes willbe popped out to enhance the property of exploration Themainmethodology of Li-RRT is based onRRTDifferent fromthe typical RRT we define a liveness set 119871 = LnLi(V) | V isin 119881initially (line (1))

At each iteration node max Li pops with max livenessfrom 119871 in line (3) Then an input vector is randomly selectedfrom functionRandU (line (5)) according towhich a randomstate in 119862free is generated Meantime 119909rand is chosen from119883goal with probability W (line (13)) It is proposed as a goalbias samplingmethodwhich is simple but effective Similarlythe nearest node is calculated given the cost function 119888(sdot)New node 119909new then is achieved based on forward processingdrive (line (20)) CollisionFree is called to verify the validityof 119909new and (119909nearest 119909new) (line (21)) Expanding process isimplemented by adding 119909new and (119909nearest 119909new) (line (22))

Liveness of each node is updated by UpdateLi(V) inAlgorithm 3 From the definition of LnLi(119909119896) (see (5)) weonly need to care about three nodes sets that is 1198811ch 1198812ch 1198813ch1198811ch contains only 119909new since Se(119909119896) is constant and isdetermined based on the state and the environments Herewe define that ldquoall parent nodesrdquo of 119909119896 are the nodes setV | 119909119896 can be traced from V different from the parentnode 119909par(119909119896) 1198812ch includes exactly all parent nodes of 119909newBecause 119909new increases the successors of V isin 1198812ch we setLnLi(V isin 1198812ch) = LnLi(V) + (119889119904(V)119899119905 minus (119889119904(V) + 1)(119899119905 + 1))Specifically we set the liveness of 119909par119909(119909new) as LnLi(119909par119909) =LnLi(119909par119909) minus 1 1198813ch is the near set of 119909new that is 1198813ch = V |119888(V 119909new) le H 119898ℎ of V isin 1198813ch will increase because of 119909newThus we define LnLi(V isin 1198813ch) = LnLi(V) minus 14 Analysis of the Algorithms

To prove the following theorems we remind the definition ofattraction sequence [35] LetA = 1198600 1198601 119860119896 be a finitesequence of sets as follows (i) For each 119860 119894 there exists a set119860 119894 sube 119861119894 sube 119862free called the basin of 119860 119894 and forall119909 isin 119860 119894minus1 and119910 isin 119860 119894 and 119911 isin 119862free 119861119894 119909 minus 119910 le 119909 minus 119911 holds (ii)forall119909 isin 119861119894 there exists an 119898 such that the sequence of action1199061 1199062 119906119898 selected by LOCAL PLANNER algorithmwill bring the point into 119860 119894 sube 119861119894 (iii) 1198600 = 119909start and119860119896 = 119883goal

An attractor region 119860 119894 like a funnel as a metaphor formotions converges into a small region in the space As shownin Figure 6 a basin 119861119894 can be viewed as a safety zonewhich ensures that a point of 119861119894 will be selected by theNEAREST NEIGHBOR and potential well which attracts thepoint into 119860 119894 Given an attraction sequence A with length 119896and letting 119901119894 = 120583(119860 119894)120583(119862free) and 119901119898 = min119901119894 we have thefollowing lemma


Require C-space 119862 Obstacle space 119862obs Initial point119909start Goal region119883goal Discrete input set 119880 = [1199061 1199062 119906119891]Maximum search steps119870

Ensure Feasible trajectory 120585(1)119881 larr 119909start 119864 larr 0G larr (119881 119864) Liveness set 119871 = Li(119909start)

Initialization(2) for 119896 = 1 119870 do(3) max Li larr argmaxVisin119881LnLi(V)(4) if Rand() lt B then(5) 119906rand larr RandU(1199061 119906119891)(6) 119909rand larr drive(max Li 119906rand) 119909nearest larr max Li(7) for each 119909near isin 119883near do(8) if 119888(119909near 119909rand) lt 119888(max Li 119909rand) then(9) 119909nearest larr 119909near(10) end if(11) end for(12) else(13) if Rand() lt W then(14) 119909rand larr 119909goal isin 119883goal(15) else(16) 119909rand larr RandProc(119862free)(17) 119909nearest larr Nearest(119909rand)(18) end if(19) end if(20) 119909new larr drive(119909nearest 119909rand)(21) if CollisionFree(119909nearest 119909rand) then(22) 119881 larr 119881 cup 119909new 119864 larr 119864 cup (119909nearest 119909new)(23) end if(24) if 119909new isin 119883goal then(25) objective = TRUE(26) end if(27) for all V isin 119881 do(28) UpdateLi(V)(29) end for(30) end for(31) return 120585 if objective = TRUE else failure

Algorithm 2 Liveliness-based RRT algorithm

Require Near set thresholdH Three nodes set1198811ch 1198812ch 1198813ch(1) LnLi(119909new) larr ln Se(119909new) + [minus119898ℎ(119909new)](2) for V isin 1198812ch do(3) LnLi(V) larr LnLi(V) + (119889119904(V)119899119905 minus (119889119904(V) + 1)(119899119905 + 1))(4) end for(5) 119909par larr parent(119909new)(6) LnLi(119909par) larr LnLi(119909par) minus 1(7) for V isin 1198813ch = Near(119909new) do(8) LnLi(V) larr LnLi(V) minus 1(9) end for

Algorithm 3 Update liveness algorithm


Bi

Ai Aiminus1

Figure 6 Attraction sequence

Lemma 3 (see [36]) Let 1198621 1198622 119862119899 be independent Pois-son trials such that for 1 le 119894 le 119899 Pr[119862119894 = 1] = 119901119894 where0 lt 119901119894 lt 1 Then for C = sum119899119894=1 119862119894 120583 = E[C] = sum119899119894=1 119901119894 and0 lt 120575 le 1

Pr [C lt (1 minus 120575) 120583] lt exp(minus12058312057522 ) (7)

Lemma 4 (see [35]) If an attraction sequence of length 119896exists for a constant 120575 the probability of basic RRT algo-rithm fails to find a trajectory after 119899 iterations are at mostexp[minus(12)(119899119901119898 minus 2119896)]

Let sequence 11990110158401 11990110158402 1199011015840119899 be an ascending order that11990110158401 le 11990110158402 le sdot sdot sdot le 1199011015840119899 where 1199011015840119894 isin 1199011 1199012 119901119899 11990110158401 = min119894119901119894and 1199011015840119899 = max119894119901119894 the following theorem can be achieved

While 119862free is partitioned into 119898 connected regionsmotion planning problem for multiple AUVs could be con-sidered as an 119898 goals version of Problem 2 A(119894) = 119860(119894)1 119860(119894)2 119860(119894)

119896119894 is an attraction sequence which connects 119909start and119883119894goal where 119896119894 is the length of attraction sequence Let 119901(119894)119895 =

120583(119860(119894)119895 )120583(119862free) 119894 = 1 2 119898 119895 = 1 2 119896119894 119901119898 =min119894119895119901(119894)119895 we have the following theorem

Theorem 5 If 119898 attraction sequences of length 119896119894 119894 =1 2 119898 exist the probability of basic RRT algorithmfails to find 119898 trajectories after 119899 iterations are at mostexp[minus(12)(119899119901119898 minus 2sum119898119894=1 119896119894)]Theorem 6 If 119898 attraction sequences of length 119896119894 119894 =1 2 119898 exist the probability of liveliness-based RRT algo-rithm fails to find 119898 trajectories after 119899 iterations are at mostexp[(1119899)sum119898119894=1sum119899119894119895=1((1199011015840(119894)119895 119901119898)(minus(12)119899119901119898 + sum119898119894=1 119896119894))]Proof Let sequence (1199011015840(119894)1 1199011015840(119894)2 1199011015840(119894)119899119894 ) be 119898 ascendingorders satisfying (1199011015840(119894)1 le 1199011015840(119894)2 le sdot sdot sdot le 1199011015840(119894)119899119894 ) where 1199011015840(119894)1 =min1198951199011015840(119894)119895 and 1199011015840(119894)119899119894 = max1198951199011015840(119894)119895 119895 = 1 2 119899119894 119894 = 1 2 119898 While using the strategy of eliminating the ldquouselessrdquonodes the random variableC is viewed as Poisson trails withprobability distribution (1199011015840119894119895 ) 119894 = 1 2 119898 119895 = 1 2 119899119894C is viewed as Poisson trails with probability distribution1199011015840(119894)119895 then 120583 = E[C] = sum119898119894=1sum119899119894119895=1 1199011015840(119894)119895 and 0 lt 120575 le 1

According to Lemma 3 Pr[C lt (1 minus 120575)120583] lt exp(minus12058312057522)where 120575 = 1 minus sum119898119894=1 119896119894(119899119901119898) In addition

minus12058312057522 = minus12119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (1 minus sum119898119894=1 119896119894119899119901119898 )2

= 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894 minus (sum119898119894=1 119896119894)22119899119901119898 )

lt 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)

(8)

This completes the proof

Proposition 7 Given specific 119862 119862free 119909start and a set of goalregions 1198831goal 1198832goal 119883119898goal while solving Problem 2the liveliness-based RRT algorithm always has smaller failingprobability than the basic RRT algorithm

Proof Note that (1119899)sum119898119894=1sum119899119894119895=1(1199011015840119894119895 119901119898) gt 1 according toTheorems 5 and 6

Pr [LiRRT] = exp[[1119899119898sum119894=1

119899119894sum119895=1

1199011015840(119894)119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)]]

lt exp[minus12 (119899119901119898 minus 2 119898sum119894=1

119896119894)]= Pr [bRRT]

(9)

In summary we can see that Li-RRT enhances theexpanding efficiency from Proposition 7

5 Applications of Li-RRT

In this section we present a numerical simulation for planarAUV with data from NOAA A region of Hawaii that isa rectangle from 157∘5810158404810158401015840W 21∘2010158402410158401015840N to 157∘5710158403610158401015840W21∘2110158403610158401015840N is utilized to testify the effectiveness of Li-RRTsee Figure 7(a) Without loss of generality we assume thatAUV only voyages at depth of 5 meters at least And thedynamics of AUV is = V cos(120579) 119910 = V sin(120579) 120579 = 120596Meantime we set 0 le |V| le 2ms and 0 le |120596| le 015 rads

We can see that a valid path is found by Li-RRT shownas in Figure 7(b) Although solutions returned by Li-RRT aremostly not optimal it can supply candidate paths effectivelyin less planning time A comparison between RRT and Li-RRT is described in Figure 8 Li-RRT obviously costs lesssearching iterations by utilizing existing information betweencurrent random tree and the environment

Here we take multiple simulations by RRT with sameconditions and present two better solutions We can seethat RRT needs more searching iterations than Li-RRTaveragely Sampling strategy in typical RRT utilizes only goalbias method which tries to make the random searching


(a) Hawaii DEM figure

Li-RRT (iteration 1026)

(b) Path found by Li-RRT

Figure 7 Start position locates at 157∘5810158403010158401015840W 21∘2010158402910158401015840N goal locates at 157∘5710158404710158401015840W 21∘2110158401410158401015840N shown as a solid blue circle Red linedescribes a valid path from start to goal

RRT (iteration 2075)

(a) RRT with 2075 iterations


(b) RRT with 3093 iterations

Figure 8 Planning with the same conditions by RRT Start position locates at 157∘5810158403010158401015840W 21∘2010158402910158401015840N Goal locates at 157∘5710158404710158401015840W21∘2110158401410158401015840N shown as a solid blue circle Red line describes a valid path from start to goal

tree towards the goal region It certainly may generateuseless sampling nodes in every searching step For examplesampling that avoids an oversampled areas seems moreefficient however typical RRT has no such ability ThusRRT needs more searching time or iterations to reach thegoal

6 Conclusions

In this paper we have presented an analysis of rapidlyexploring random trees and defined the several indicators of atree including the coverage and the growth speed describingthe growing behavior of a tree Considering the externaland internal factors which influence the expanding ability

we also have defined indicators collision detective indexdegree offspring and neighbor collisions On this basiswe have proposed Li-RRT for AUVs motion planning Byusing the tools attraction sequence theoretical analysis hasindicated that liveliness-based RRT enhances the expandingspeed Simulations are also provided to show the effectivenessof Li-RRT It has been shown that Li-RRT performs wellin finite planning time Moreover the growth direction ofa tree also reflects some properties of a treersquos expandingability which could be utilized by RRT to conduct theexpanding procedure One of the possible research directionswill focus on the planning without any prior informationin the uncertain environment considering such indicatorsproposed in this work


Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (NSFC) under Grants 61472325 and61273333

References

[1] Y J Heo and W K Chung ldquoRRT-based path planning withkinematic constraints of AUV in underwater structured envi-ronmentrdquo in Proceedings of the 10th International Conference onUbiquitous Robots and Ambient Intelligence URAI rsquo13 pp 523ndash525 November 2013

[2] A Gasparetto P Boscariol A Lanzutti and R VidonildquoPath planning and trajectory planning algorithms a generaloverviewrdquo Mechanisms and Machine Science vol 29 pp 3ndash272015

[3] O Khatib ldquoReal-time obstacle avoidance for manipulatorsand mobile robotsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo85) vol 2 pp500ndash505 1985

[4] S S Ge and Y J Cui ldquoNew potential functions for mobile robotpath planningrdquo IEEE Transactions on Robotics and Automationvol 16 no 5 pp 615ndash620 2000

[5] CM Saaj V Lappas andVGazi ldquoSpacecraft swarmnavigationand control using artificial potential field and slidingmode con-trolrdquo in Proceedings of the 2006 IEEE International Conferenceon Industrial Technology ICIT rsquo06 pp 2646ndash2651 2007

[6] P F J Lermusiaux T Lolla P J H et al Science of AutonomyTime-Optimal Path Planning and Adaptive Sampling for Swarmsof Ocean Vehicles Springer International Publishing Gewerbe-strasse Switzerland 2016

[7] S Carpin and G Pillonetto ldquoMerging the adaptive randomwalks planner with the randomized potential field plannerrdquoin Proceedings of the in Proceedings of the Fifth InternationalWorkshop on RobotMotion and Control 2005 pp 151ndash156 2005

[8] S A Wilmarth N M Amato and P F Stiller ldquoMAPRM aprobabilistic roadmap plannerwith sampling on themedial axisof the spacerdquo inProceedings of the IEEE International Conferenceon Robotics and Automation pp 1024ndash1031 May 1999

[9] L Han and N M Amato ldquoA kinematics-based probabilisticroadmap method for high DOF closed chain systemsrdquo IEEEInternational Conference on Robotics and Automation vol 1 pp473ndash478 2004

[10] V Boor M H Overmars and A F V D Stappen ldquoGaussiansampling strategy for probabilistic roadmap plannersrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsand Automation ICRA rsquo99 pp 1018ndash1023 May 1999

[11] R Cui B Gao and J Guo ldquoPareto-optimal coordination ofmultiple robots with safety guaranteesrdquo Autonomous Robotsvol 32 no 3 pp 189ndash205 2012

[12] S M LaValle ldquoRapidly-exploring random trees a new tool forpath planningrdquo Tech Rep 98-11 Computer Science Dept IowaState University Iowa Iowa USA 1998

[13] J Kim and J P Ostrowski ldquoMotion planning a aerial robot usingrapidly-exploring random trees with dynamic constraintsrdquo in

Proceedings of the in IEEE International Conference on Roboticsand Automation 2003 vol 2 pp 2200ndash2205 2003

[14] D Shim H Chung H J Kim and S Sastry ldquoAutonomousexploration in unknown urban environments for unmannedaerial vehiclesrdquo in Proceedings of the in AIAA GNC Conferencepp 1ndash8 2005

[15] R Cui Y Li andW Yan ldquoMutual information-basedmulti-auvpath planning for scalar field sampling using multidimensionalRRTrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 46 no 7 pp 993ndash1004 2016

[16] T Van Huynh M Dunbabin and R N Smith ldquoConvergence-guaranteed time-varying RRT path planning for profiling floatsin 4-dimensional flowrdquo in Proceedings of the AustralasianConference on Robotics and Automation ACRA rsquo14 December2014

[17] J L Blanco M Bellone and A Gimenez-Fernandez ldquoTP-spaceRRT - Kinematic path planning of non-holonomic any-shapevehiclesrdquo International Journal of AdvancedRobotic Systems vol12 article no A55 2015

[18] M Kalisiak and M van de Panne ldquoRRT-blossom RRT witha local flood-fill behaviorrdquo in Proceedings of the IEEE Interna-tional Conference on Robotics and Automation (ICRA rsquo06) pp1237ndash1242 May 2006

[19] J M Esposito J Kim and V Kumar Adaptive RRTs forValidating Hybrid Robotic Control Systems Springer BerlinGermany 2005

[20] J Si and C Chin ldquoAn adaptable walking-skid for seabed ROVunder strong current disturbancerdquo Journal of Marine Scienceand Application vol 13 no 3 pp 305ndash314 2014

[21] R Cui L Chen C Yang and M Chen ldquoExtended stateobserver-based integral slidingmode control for an underwaterrobot with unknown disturbances and uncertain nonlineari-tiesrdquo IEEE Transactions on Industrial Electronics vol 64 no 8pp 6785ndash6795 2017

[22] M D Thekkedan C S Chin and W L Woo ldquoVirtual realitysimulation of fuzzy-logic control during underwater dynamicpositioningrdquo Journal of Marine Science and Application vol 14no 1 pp 14ndash24 2015

[23] C S Chin W P Lin and J Y Lin ldquoExperimental validationof open-frame rov model for virtual reality simulation andcontrolrdquo Journal of Marine Science Technology vol no 2 p 212017

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

[25] O Salzman andDHalperin ldquoAsymptotically near-optimal RRTfor fast high-quality motion planningrdquo IEEE Transactions onRobotics vol 32 no 3 pp 473ndash483 2013

[26] M Otte and E Frazzoli ldquoRRTX Asymptotically optimal single-query sampling-basedmotion planningwith quick replanningrdquoInternational Journal of Robotics Research vol 35 no 7 pp 797ndash822 2016

[27] Y Li Z Littlefield and K E Bekris ldquoSparse methods for effi-cient asymptotically optimal kinodynamic planningrdquo SpringerTracts in Advanced Robotics vol 107 pp 263ndash282 2015

[28] D J Webb and J Van Den Berg ldquoKinodynamic RRTlowastAsymptotically optimal motion planning for robots with lineardynamicsrdquo Computer Science 2012

[29] P Pharpatara B Herisse and Y Bestaoui ldquo3-D trajectoryplanning of aerial vehicles using RRTlowastrdquo IEEE Transactions onControl Systems Technology vol 25 no 3 pp 1116ndash1123 2017


[30] D Lee H Song and D H Shim ldquoOptimal path planningbased on spline-RRTlowast for fixed-wing UAVs operating in three-dimensional environmentsrdquo in Proceedings of the 14th Interna-tional Conference on Control Automation and Systems ICCASrsquo14 pp 835ndash839 October 2014

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

[32] V Boor M H Overmars and A F V D Stappen GaussianSampling for Probabilistic Roadmap Planners Utrecht Univer-sity Thenetherlands Utrecht Netherlands 2001

[33] S Karaman and E Frazzoli ldquoSampling-based motion planningwith deterministic 120583-calculus specificationsrdquo in Proceedings ofthe 48th IEEE Conference on Decision and Control pp 2222ndash2229 December 2009

[34] J M Esposito J Kim and V Kumar ldquoAdaptive RRTs forValidating Hybrid Robotic Control Systemsrdquo in AlgorithmicFoundations of Robotics VI vol 17 of Springer Tracts inAdvancedRobotics pp 107ndash121 Springer Berlin Heidelberg Berlin Hei-delberg 2005

[35] S M LaValle and J J Kuffner Jr ldquoRandomized kinodynamicplanningrdquo International Journal of Robotics Research vol 20 no5 pp 378ndash400 2001

[36] R Motwani Randomized Algorithms Cambridge UniversityPress Cambridge UK 1995

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Particularly each edge is kinematically feasible once it isgenerated by two points in TP-space In [18] the expandingprocess utilizes implicit flood-fill-like mechanism to avoidthe local minimumThus the expected time to find the solu-tion is reduced Because of the insensitivity to the dimensionof problems it is able to be used to search time invariantparameters [19] It is a fact that AUVs have been widelyapplied in the marine resources exploration and utilization[20ndash23] In order to map a scalar field mutual informationbased multidimensional RRT algorithm was presented formultiple AUVs in [15]

However RRT is proven to converge almost to a nonop-timal path [24] in which RRTlowast was detailed to be asymptoti-cally optimal That means RRTlowast will ldquoalmost surelyrdquo find theoptimal path as the number of samples tends to infinity RRTlowastapplies the reconstruction of the searching tree which alwayscontains the optimal subtree at each iteration Since then sev-eral RRTlowast based optimal planning algorithms were proposed[25ndash28] and applied in many fields [29 30] LBT-RRT wasproposed in [25] which is a combination algorithmwith RRTand RRTlowast It defines a variable 120598 to describe the ldquodistancerdquoaway from the optimal path In other words 120598 can be viewedas the comparison between RRT and RRTlowast If 120598 equals zeroit becomes RRTlowast essentially In [26] RRTlowast is refined tobe capable of replanning with dynamic environments In[27] the limitation that is requirements for an asymptoticaloptimal steer function during the treersquos expansion is provablyaddressed based on sparse data structure

From [31] we know that the sampling strategies havemuch effect on the efficiency of sampling-based algorithmsIn early researches typical choice is sampling uniformly inC-space [8 12 24] Every node in C-space is sampled withexactly the same probability by which the prior informationactually is not exploited For instance a new node intuitivelyprefers to be sampled away from the region where a largenumber of samples already exist And it certainly reduces theprobability that samples are generated in the case of narrowpassages [31] For this reason various heuristic methodolo-gies are designed Goal bias is generally utilized to enhanceextending towards the goal due to its effectiveness Medialaxis is combined with path planning algorithms to searchthe C-space with much more efficiency [8] On the contraryGaussian sampling strategy prefers the region nearby obsta-cles [10 32] In this view geometric shapes of obstacles areable to be quickly constructed Some other hybrid samplingstrategies are proposed to combine two or more methodsThese methods uniformly consider the information of theenvironment to refine the sampling probability howeverthe information of nodes is not totally utilized Specificallyregions where a number of nodes are already sampled canbe mostly considered with less worth In other words it ispreferred to sample in ldquonewrdquo regions

In this paper we demonstrate the expanding mechanismof the random tree and make an estimation of the expandingability of each node We present several new indicators inthe expanding procedure which are utilized in our new algo-rithm Also we detail the indicators quantitative descriptionsof the nodes Meanwhile we provide theoretical analysisand property of our algorithm with mathematical proof (ie

probabilistic completeness and outperforming compared toRRT) The proof is based on a technical notation attractionsequence and probability theory

The reminder of the paper is organized as followsSection 2 introduces the definition and notations in thispaper The indicators which reflect the growing of the treeand expanding ability are analyzed in Section 3 Then ournew algorithm that is Li-RRT is described in detail InSection 4 the analysis of Li-RRT is provided Section 5presents an application example for a planar mobile robot Infinal concluding remarks are drawn in Section 6

2 Preliminaries and Problem Formation

21 Problem Formulation Path planning problem can begenerally viewed as a search problem in the C-space 119862 sube R119899For instance the C-space of a planar two-wheel mobile robotcan be defined as 119862 sube R3 that is position (119909 119910) headingangle 120579 In this way any 119909 isin 119862 sube R3 can describe thestate of the mobile robot Additionally we define 119862obs sub 119862and 119862free = 119862 119862obs as the obstacle C-space and free C-space separately The path planning problems require a finitesequence of states in 119862free from an initial state to a given goalstate Here we present the basic path planning problem asdefined in [24]

Problem 1 (path planning problem [24]) The bounded C-space 119862 obstacle space 119862obs initial state 119909start isin 119862free goalregion 119883goal isin 119862free denoted as tuple 119909start 119862free 119883goalreturn a path 120585 [0 1] rarr 119862free such that 120585(0) = 119909start120585(1) isin 119883goal If there exist no feasible paths return failure

Facing path planning problems each element of dimen-sion is considered independently with no other constraintsHowever it is unrealistic inmost applications Specifically thekinodynamic of AUVs is restricted in case of uncontrollabil-ity It means that there exists corresponding available inputsequence 119906(119904) [0 1] rarr 119880 that drives the AUV from 119909start to119883goal Subsequently the path can also be viewed as a functionof input sequence 120585(119906(sdot)) 119906(sdot) rarr 119862free Formally we give thedefinition of trajectory planning problem for AUVs

Problem 2 (trajectory planning problem for AUVs) Given119909start 119862free 119883goal and kinodynamic constraint of AUVs =119891(119909 119906) return an available control sequence 119906 [0 1] rarr 119880such that the path 120585(119906(sdot)) = 119909 isin 119862free | 120585(119906(0)) = 119909init120585(119906(1)) isin 119862free 120585(119906(119904)) = int1

0[119909init + 119891(119909 119906)]119889119904

In this paper we focus on the trajectory planning problemfor AUVs Since the initial force has major impact on themotion of AUVs compared to ground vehicles the kinody-namic constraints are more complex

3 Algorithms

RRT was firstly introduced by Lavalle as a sampling-basedpath planning algorithm which is shown in Algorithm 1It has the ability of quickly searching or exploring the C-space by a random tree At each iteration the tree expands


Search tree

Grid points

Node

Grid space

ＧＣＨ(dj )






119888 (G) = 1120575119899119892sum119895=1

min (119889119895 120575)119899119892 (1)




119892 (G) = minusΔ119888 (G)Δ119899119905 (2)





xk

u1

u2

u3

un

x1k+1

x2k+1

x3k+1

xnk+1









Se (G) = sum119899119897119896=1

Se (119909119896)119899119897 (3)







119889119904 (119909119896) = sum119909119894isinUs(119909119896)

119889 (119909119894) (4)













d(x3) = 0

ds(x3) = 0

d(x6) = 0

ds(x6) = 0

d(x2) = 3

ds(x2) = 3

3

6d(x7) = 0

ds(x7) = 0

7

d(x1) = 2

ds(x1) = 6

d(x4) = 1

ds(x4) = 1

d(x5) = 0

ds(x5) = 0

1

4

5

2


Obstacle Obstacle

Obstacle

xk

u1u2

un

x1kx2

k

xnk















Bi

Ai Aiminus1











minus12058312057522 = minus12119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (1 minus sum119898119894=1 119896119894119899119901119898 )2

= 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894 minus (sum119898119894=1 119896119894)22119899119901119898 )

lt 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)

(8)




Pr [LiRRT] = exp[[1119899119898sum119894=1

119899119894sum119895=1

1199011015840(119894)119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)]]


119896119894)]= Pr [bRRT]

(9)

















6 Conclusions






Acknowledgments


References







































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Search tree

Grid points

Node

Grid space

ＧＣＨ(dj )






119888 (G) = 1120575119899119892sum119895=1

min (119889119895 120575)119899119892 (1)




119892 (G) = minusΔ119888 (G)Δ119899119905 (2)





xk

u1

u2

u3

un

x1k+1

x2k+1

x3k+1

xnk+1









Se (G) = sum119899119897119896=1

Se (119909119896)119899119897 (3)







119889119904 (119909119896) = sum119909119894isinUs(119909119896)

119889 (119909119894) (4)













d(x3) = 0

ds(x3) = 0

d(x6) = 0

ds(x6) = 0

d(x2) = 3

ds(x2) = 3

3

6d(x7) = 0

ds(x7) = 0

7

d(x1) = 2

ds(x1) = 6

d(x4) = 1

ds(x4) = 1

d(x5) = 0

ds(x5) = 0

1

4

5

2


Obstacle Obstacle

Obstacle

xk

u1u2

un

x1kx2

k

xnk















Bi

Ai Aiminus1











minus12058312057522 = minus12119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (1 minus sum119898119894=1 119896119894119899119901119898 )2

= 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894 minus (sum119898119894=1 119896119894)22119899119901119898 )

lt 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)

(8)




Pr [LiRRT] = exp[[1119899119898sum119894=1

119899119894sum119895=1

1199011015840(119894)119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)]]


119896119894)]= Pr [bRRT]

(9)

















6 Conclusions






Acknowledgments


References







































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










xk

u1

u2

u3

un

x1k+1

x2k+1

x3k+1

xnk+1









Se (G) = sum119899119897119896=1

Se (119909119896)119899119897 (3)







119889119904 (119909119896) = sum119909119894isinUs(119909119896)

119889 (119909119894) (4)













d(x3) = 0

ds(x3) = 0

d(x6) = 0

ds(x6) = 0

d(x2) = 3

ds(x2) = 3

3

6d(x7) = 0

ds(x7) = 0

7

d(x1) = 2

ds(x1) = 6

d(x4) = 1

ds(x4) = 1

d(x5) = 0

ds(x5) = 0

1

4

5

2


Obstacle Obstacle

Obstacle

xk

u1u2

un

x1kx2

k

xnk















Bi

Ai Aiminus1











minus12058312057522 = minus12119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (1 minus sum119898119894=1 119896119894119899119901119898 )2

= 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894 minus (sum119898119894=1 119896119894)22119899119901119898 )

lt 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)

(8)




Pr [LiRRT] = exp[[1119899119898sum119894=1

119899119894sum119895=1

1199011015840(119894)119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)]]


119896119894)]= Pr [bRRT]

(9)

















6 Conclusions






Acknowledgments


References







































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












d(x3) = 0

ds(x3) = 0

d(x6) = 0

ds(x6) = 0

d(x2) = 3

ds(x2) = 3

3

6d(x7) = 0

ds(x7) = 0

7

d(x1) = 2

ds(x1) = 6

d(x4) = 1

ds(x4) = 1

d(x5) = 0

ds(x5) = 0

1

4

5

2


Obstacle Obstacle

Obstacle

xk

u1u2

un

x1kx2

k

xnk















Bi

Ai Aiminus1











minus12058312057522 = minus12119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (1 minus sum119898119894=1 119896119894119899119901119898 )2

= 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894 minus (sum119898119894=1 119896119894)22119899119901119898 )

lt 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)

(8)




Pr [LiRRT] = exp[[1119899119898sum119894=1

119899119894sum119895=1

1199011015840(119894)119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)]]


119896119894)]= Pr [bRRT]

(9)

















6 Conclusions






Acknowledgments


References







































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















Bi

Ai Aiminus1











minus12058312057522 = minus12119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (1 minus sum119898119894=1 119896119894119899119901119898 )2

= 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894 minus (sum119898119894=1 119896119894)22119899119901119898 )

lt 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)

(8)




Pr [LiRRT] = exp[[1119899119898sum119894=1

119899119894sum119895=1

1199011015840(119894)119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)]]


119896119894)]= Pr [bRRT]

(9)

















6 Conclusions






Acknowledgments


References







































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Bi

Ai Aiminus1











minus12058312057522 = minus12119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (1 minus sum119898119894=1 119896119894119899119901119898 )2

= 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894 minus (sum119898119894=1 119896119894)22119899119901119898 )

lt 1119899119898sum119894=1

119899119894sum119895=1

1199011015840119894119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)

(8)




Pr [LiRRT] = exp[[1119899119898sum119894=1

119899119894sum119895=1

1199011015840(119894)119895119901119898 (minus12119899119901119898 + 119898sum119894=1

119896119894)]]


119896119894)]= Pr [bRRT]

(9)

















6 Conclusions






Acknowledgments


References







































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















6 Conclusions






Acknowledgments


References







































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Acknowledgments


References







































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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Liveness-Based RRT Algorithm for Autonomous …downloads.hindawi.com/journals/jat/2017/7816263.pdfpath planning algorithm, which is shown in Algorithm 1. It has the ability of quickly