Expert Systems with Applications 36 (2009) 4688–4695

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Expert Systems with Applications

journal homepage: www.elsevier .com/locate /eswa

Motion planning in order to optimize the length and clearance applying aHopfield neural network

Mehdi Ghatee a,b,*, Ali Mohades a,c

a Department of Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran 15875-4413, Iranb Laboratory of Network and Optimization Research Center (NORC), Tehran, Iranc Laboratory of Algorithms and Computational Geometry Group, Tehran, Iran

a r t i c l e i n f o a b s t r a c t

Keywords:Multi-objectiveOnline routingNeural networkParallel implementation

0957-4174/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.eswa.2008.06.040

* Corresponding author. Address: Department of CUniversity of Technology, No. 424, Hafez Ave., Tehran 164542542; fax: +98 21 66497930.

E-mail addresses: ghatee@aut.ac.ir (M. Ghatee),hades).

This paper deals with motion planning in plane for a mobile robot with two freedom degrees throughsome polygonal unmoved obstacles. Applying Minkowski sum, we can represent the robot as a point.Then, by using traditional approaches such as visibility graphs, simple and generalized Voronoi diagrams,decomposition methods, etc, it is possible to provide a graph covering obstacles, say roadmap. In order tofind a real-time collision-free robot motion planning between two arbitrary source and target configura-tions through the roadmap, an adoptive Hopfield neural network is considered. Maximizing the clearanceof path together with minimizing the length of path are pursued in a bi-objective framework. For treatingwith multiple objectives TOPSIS method, as a kind of goal programming techniques, is provided to findthe efficient solutions. Because of capability of parallel computation through hardware implementationof neural networks, the presented approach is a reasonable technique in mobile robot navigation andtraveler guidance systems. The advantages of the proposed system are confirmed by simulation experi-ments. This approach can be directly extended in unknown environment including time-varyingconditions.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Over the last decade there has been considerable progress inmotion planning techniques and their application (Ahuactzin &Gupta, 1998; Chou, Chou, & Chen, 2008). Classic path planning ap-proaches including roadmap, cell decomposition and potentialfield use global methods to search the possible paths in the work-space. These models deal with static environment only and arecomputationally complicated. This paper considers a robot withtwo freedom degrees in the plane among polygonal obstacles,i.e., the robot can only transformed not rotate in the plane. Motionplanning consist of a natural looking collision-free path for a robot.This means the path should be short as well as it should have aguaranteed amount of clearance, that is any point on the path ispossibly far from the closest obstacle. In Wein, van den Berg, andHalperin (2007) some other objectives such as smoothness andnot containing any sharp turns, were also introduced. Usually thereis some confliction between these objective functions, for instance

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omputer Science, Amirkabir5875-4413, Iran. Tel.: +98 21

mohades@aut.ca.ir (A. Mo-

it is possible to considerably shorten the path by taking a shortcutthrough a narrow passage.

The motion planning problem can be efficiently solved by com-puting a complete representation of the free configuration space.This approach was simplified, by decomposing the configurationspace into pseudo-trapezoidal cells and constructing a roadmapof the free cells (Wein et al., 2007). Another popular approach isto use Probabilistic Roadmaps (Kavraki, Svestka, Latombe, & Over-mars, 1996). But the output paths in this case are also piecewiselinear and may be far from the shortest possible paths. As the infra-structure of motion route design, some methods can be pursued inorder to create a graph considering the place of obstacles (Berg, vanKreveld, Overmars, & Schwarzkopf, 2000; LaValle, 2006; Plaku,Bekris, Chen, Ladd, & Kavraki, 2005). After this preprocessing, aquery phase should be done to connect the source and target con-figurations through the edges of the provided graph. In this phase asingle objective or multiple objectives may be considered (Bu &Cameron, 2002; Min, Zhu, & Zheng, 2005). Most of the techniquesimplemented for this aim, generate locally optimal paths not nec-essarily with online response. To provide online shortest path inSadati and Taheri (2002) a neural network architecture is devel-oped. Some neural network models were proposed for realtime ro-bot motion planning through learning and obtaining dynamicnavigation of a mobile robot with obstacle avoidance, however,

Fig. 1. Using decomposition method to create roadmap.

Fig. 2. Approximate decomposition into rectangles.

M. Ghatee, A. Mohades / Expert Systems with Applications 36 (2009) 4688–4695 4689

the result of these methods are not optimal, particularly at theinitial learning phase (Glasius, Komoda, & Gielen, 1995). Thus, aHopfield-type neural network model is more reasonable for dy-namic trajectory formation without learning.

After the concept of Hopfield network in traveling salesmanproblem has been initially proposed by Hopfield and Tank(1985), solving routing problem by this approach was pursued(Smeda & El-Hawary, 1999; Xia & Wang, 2000; Araújo, Ribeiro, &Rodrigues, 2001; Venkataram, Ghosal, & Kumar, 2002). Findingapproximate Pareto solutions for multi-objective optimizationproblem, by help of Hopfield network, was also addressed inBalakrishnan, Kannan, Aravindan, and Subathra (2003). Because,Hopfield networks and its variants offer inherent capability tomassively parallelize the computations required to search forsolutions of large-scale optimization problems, they provideappropriate efficiency (Chichocki & Unbehauen, 1993; Shen &Wang, 2008).

In this paper a new path planning scheme is addressed to es-cape from local minimum. This idea can be also generalized for un-known environments. Note that in Burago, Grigoriev, and Slissenko(2004) this problem has been solved in 3-dimension space consid-ering one objective function not online. The approach of this paperis similar to that of Yang and Meng (2000) and a neural network isused to real-time shortest path as well as collision-free motionplanning. The Pareto optimal solution which will be obtained fromthis approach does a trade off between (a) the shortest path lengthsubject to obstacle imposed constraints and (b) the obstacle clear-ance, which is defined as the minimum distance between any pointon the vehicle and any obstacle, at any time during the path tra-versal. The rest of paper is organized as follows:

In the next section some preliminaries are introduced. In Sec-tion 3, the Hopfield approach is addressed. Section 4 shows thepossibility of using neural network in order to find Pareto optimalsolution of a multi-objective shortest path problem. The applica-tion of this idea in motion planning becomes in Section 5. Section6 ends this paper with a brief conclusion.

2. Preliminaries

2.1. Roadmap graph

The main data structure that will be utilized throughout the pa-per is the notion of a roadmap. A roadmap is a graph G = (V,E), inwhich V includes the nodes and E consists of edges. As the infra-structure of motion route design, there are some methods to createa graph with respect to the location of obstacles. Some of them, butnot all, are as follows:

1. Probabilistic roadmap construction methods (Pisula, Hoff, Lin, &Manocha, 2000),

2. Rapidly-exploring Random Tree (RRT) (Lavalle, 1998),3. Visibility graphs (Wein et al., 2007),4. Voronoi diagrams of polygons (Berg et al., 2000),5. The visibility-Voronoi diagram (Wein et al., 2007),6. Decomposition methods (Chazelle, 1987).

In this paper, we apply decomposition methods. According tothese approaches, the space C is represented as a collection of dis-joint cells and the planning is done between configurations in thesame build connectivity graph representing adjacency relations be-tween cells. Note that cells are adjacent if one can move directlybetween them (Chazelle, 1987), Fig. 1. An interesting approachfor providing connectivity graph is to use midpoints of cell bound-aries as crossing points between cells. Trapezoidal decompositionby applying a sweepline algorithm, is another powerful scheme

which can be used to decompose the space. Note that generally,it is NP-hard to compute an optimal convex decomposition of apolygon. Thus, an approximate decomposition into rectangles canbe pursued for providing connectivity graph, see Fig. 2. For conve-nience in present paper we use a uniform mesh to create the graphof roadmap. The details is presented in Section 4.

2.2. Hopfield architecture for optimization

Neural networks and particularly Hopfield networks and itsvariants offer inherent capability to massively parallelize the com-putations required to search for solutions of large-scale optimiza-tion problems. This computational promise is likely to bedelivered if hardware realization of these neural network algo-rithms becomes a reality. However, when the hardware implemen-tation becomes infeasible, simulation of the Hopfield networkalgorithm on Von Neumann architectures, including those offeringlarge-grain parallelism, is the only viable option to explore theircomputational potential (Serpen, 2004). Therefore, we only dem-onstrate and validate theoretical inferences with respect to theircomputational promise.

Among a lot of published references on Hopfield neural networkthis paper we adapt this concept from (Smith, Palaniswami, &Krishnamoorthy, 1998). The Hopfield (1982), Hopfield and Tank,1985 comprises a fully interconnected system of n neurons, seeFig. 3. Neuron i has internal state ui and output level vi (boundedby zero and one). The internal state ui incorporates a bias current(or negative threshold) denoted by Ii and the weighted sums ofoutputs from all other neurons. The weights, which determinethe strength of the connections from neuron j to i, are given bywi,j. The relationship between the internal state of a neuron andits output level is determined by an activation function wi (ui)which is bounded bellow by zero and above by one. Commonly,the following sigmoid function is used as activation function:

Fig. 3. Hopfield network with unsupervised learning algorithm.

4690 M. Ghatee, A. Mohades / Expert Systems with Applications 36 (2009) 4688–4695

vi ¼ wiðuiÞ ¼1

1þ expð�kuiÞ¼ 1

21þ tanh

k2

ui

� �� �; ð1Þ

where k = 1/T is a parameter used to control the gain (or slope) ofthe activation function.

Note that this function is monotonically increasing and contin-uously differentiable. Moreover, its derivative can be calculatedwith respect to itself values, because

dvi

dui¼ wiðuiÞ0 ¼ kwiðuiÞð1� wiðuiÞÞ ¼ kvið1� viÞ:

Hopfield (1982) showed that the system for hardware implementa-tion is demonstrated by the following resistance–capacitanceequations:

dui

dt¼X

j

Wi;jvj �ui

sþ Ii; ð2Þ

ui ¼ w�1i ðviÞ: ð3Þ

For high-gain activation functions (k ?1), the output values ap-proach either zero or one, and the state space of the network out-puts is the set of corners of the n-dimensional hypercube {0,1}n.The final state of the network is, therefore, one of these corners.

Hopfield networks can be used as an approximated method forsolving zero–one optimization problem, because by symmetricweights (Wi,j = Wj,i, the Eqs. (2) and (3) converge to a minimumof the following energy function:

EðvÞ ¼ �1=2X

i

Xj

Wi;jvivj �X

i

Ii:vi: ð4Þ

Thus, if a combinatorial optimization problem can be expressed interms of a quadratic energy function of the general form given by(4), a Hopfield network can be used to find locally optimal solutionsof the energy function, which may translate to a local minimumsolution of the optimization problem, see Hopfield & Tank (1985).Typically, the network energy function is made equivalent to theobjective function which is to be minimized, while each of the con-straint of the optimization problem is included in the energy func-tion as penalty terms. Clearly, a constrained minimum of theoptimization problem will also optimize the energy function, since,the objective function term will be minimized and constraint satis-faction implies that the penalty terms will be zero. Now for a givenobjective function E, we can easily obtain the following parameters:

Wi;j ¼ �o2E

ovjovi; ð5Þ

Ii ¼ �oEovi�X

j

Wi;jvj; ð6Þ

which shows that in order to optimize E, it is not necessary to re-shape objective function to the standard quadratic programmingformulation (4).

Consider the following time derivative of energy function E(v):

dEdt¼Xn

i¼1

oEovi

ovi

ot¼Xn

i¼1

�sidui

dt� aiui

� �dvi

dt

¼ �Xn

i¼1

sidvi

dui

dui

dt

� �2 !

þ �Xn

i¼1

aiuidvi

dt

!: ð7Þ

The second term sometimes increases the energy function. To avoidthis effect, the following modified differential equations areemployed:

dui

dt¼ �li

oEðvÞovi

; ð8Þ

or equivalently by eliminating internal potential ui, we get:

dvi

dt¼ �likvið1� viÞ

oEðvÞovi

; ð9Þ

The following iterative scheme converges to the local minima of en-ergy function (4):

dvidt ðnÞ ¼ �likviðnÞð1� viðnÞÞ oEðvÞ

oviðnÞ

¼ �likviðnÞð1� viðnÞÞP

jWi;jvjðnÞ þ Ii

!;

viðnþ 1Þ ¼ viðnÞ þ dvidt ðnÞ;

8>>>>><>>>>>:

ð10Þ

where li > 0 is the step size procedure or the learning coefficient. Analgorithmic implementation of these concepts may be followed inSerpen (2004).

Unfortunately, a minimum of the energy function does not nec-essarily correspond to a constrained minimum of the objectivefunction due to the fact that there are likely several terms in theenergy function which contribute to many local minima. Thus, atradeoff exists between which terms will be minimized com-pletely. Also the feasibility of the network is dependent on the pen-alty parameters and so they should be chosen carefully.Furthermore, even if the network converges to a feasible solution,its quality is likely to be poor compared to other techniques, sincethe Hopfield network is a descent technique and converges to thefirst local minimum which it encounters. An promising approachwhich frequently exercised is the use of simulated annealing strat-egy for searching the state of neurons corresponding to the globalminimum of the energy function (Chichocki & Unbehauen, 1993, p.114). With respect to this idea, the following iterative differentialequation is utilized instead of (9) which considers artificial thermalnoise which is gradually decreased in time. This noise allows occa-sional hill-climbing interspersed with descents:

dvi

dt¼ �lik við1� viÞ

oEðvÞovi

þ cðtÞNi

� �

¼ �lik við1� viÞX

j

Wi;jvj þ Ii

!þ cðtÞNi

" #; ð11Þ

where Ni is zero-mean uncorrelated noise sources and c(t) is theannealing schedule (typically c(t) = c0/exp(�t/s)). Decreasing thevariance of the noise during the optimization process is similar toperforming stochastic annealing (Chichocki & Unbehauen, 1993, p.486).

M. Ghatee, A. Mohades / Expert Systems with Applications 36 (2009) 4688–4695 4691

3. Multi-objective shortest path problem

Multi-objective optimization problems deal with the presenceof different conflicting objectives. Given that it is not possible toobtain a single solution by optimizing all the objectives simulta-neously, a common way to face these problems is to obtain a setof efficient solutions called the non-dominated or Pareto optimalsolutions. In motion planning models or route guidance systems,multi-objective shortest path problem has an essential role (Spais& Petron, 2003). This problem concentrates on finding the pathwith minimum distance, time or cost from a source node to a tar-get node. Sometimes, this program has to be solved in real time inwhich artificial neural networks (ANN) (Chichocki & Unbehauen,1993) appears as an ideal candidate instead of deterministic algo-rithms. A well-known ANN for solving optimization problems isHopfield network which has been proposed to solve travelingsalesman problem by Hopfield & Tank (1985). ANNs process datain parallel, thus, they speed up the computation. A data costlieststructure for simulation of Hopfield neural network algorithm forlarge scale optimization problems was implemented in Serpen(2004). In Cavalieri & Russo (1998) a method by fuzzy logic-basedcoefficient tuning for improving Hopfield neural network perfor-mance was introduced. Also in Wang, Tang, & Cao (2002) a simplelearning method in Hopfield neural network for combinatorialoptimization problems was addressed. An adaptive Hopfield neuralnetwork for finding approximate Pareto solutions for multi-objec-tive optimization problem was provided in Balakrishnan et al.(2003). Furthermore, Hopfield neural network in routing has beenapplied in Araújo et al. (2001), Hákkinen, Lagerholm, Peterson, &Sóderberg (2000), Smeda & El-Hawary, 1999, Venkataram et al.,2002, Xi & Wang (2000). In Kurokawa, Ho, & Mori (1998) a paralleldecentralized scheme in place of large size Hopfield type neuralnetwork was discussed which can be used in network routing.Our aim is to extend Hopfield neural network in order to find on-line multi-objective shortest path. In what follows, the mathemat-ical formulation of this problem together with Hopfield networkimplementation will be presented.

3.1. Mathematical formulation

Let G = (V,E) be a directed graph in which V = {1,. . .,n}, and E arethe sets of nodes and edges, respectively. Assign to each edge(i,j) 2 E, the vector costs cp

i;j; p = 1,. . .,P. Let s,d 2 N be the sourceand target nodes. The multi-objective shortest path problem be-tween s and d can be expressed as the following zero–one integerlinear programing problem:

minimize zp ¼Xði;jÞ2A

cpi;jxi;j; p ¼ 1; . . . ; P; ð12Þ

s.t.

Xj:ði;jÞ2A

xi;j �X

j:ðj;iÞ2A

xj;i ¼1 i ¼ s;

�1 i ¼ d;

0 i–s;d;

8><>: ð13Þ

xi;j 2 f0;1g; ð14Þ

where xi,j = 1 if and only if the edge (i,j) is shared in path joining sand d.

To simplify, hereafter, denote the optimal value of pth objectivefunction with zp

� : It is also possible to get these quantities as deci-sion maker’s goals. Using TOPSIS method (Hwang & Yoon, 1981), asone kind of goal programming schemes, permits to define the fol-lowing weighed combination of differences between objectivesand their targets which can be utilized in place of P disjoint objec-tive functions (12):

minimizeXP

p¼1

hp

Xði;jÞ2A

cpi;j:xi;j � zp

�

!2

; ð15Þ

where hp P 0 is the importance weight of the pth objective functionin decision making process. As much as each term of objective func-tion (15) approaches zero, the obtained path is close to the optimalsolution of the shortest path problem with respect to the corre-sponding objective function.

Now define the following strategy function for multi-objectiveshortest path problem similar to that of Venkataram et al. (2002)utilizing fixed positive parameters a1, a2, a3, a4, a5, and a6,

EðvÞ ¼PPp¼1

hpPni¼1

Pnj¼1

cpi;j:xi;j � zp

�

!2

þ a1Pni¼1

Pnj¼1

ci;j � xi;j

þa2P

i–s;d

Pnj¼1

xi;j �Pnj¼1

xj;i

!2

þa3Pnj¼1

xs;j � 1

!2

þ a4Pnj¼1

xj;d � 1

!2

þa5ð1� xd;sÞ þ a6Pni¼1

Pnj¼1

xi;jð1� xi;jÞ;

ð16Þ

where ci,j = 1 if and only if an edge from node i to node j is not exist,otherwise ci,j = 0.

The justification of choosing this strategy function is givenbelow:

� The first term is the objective function (15) to minimize the totalcost of a path by taking into account the multiple costs of theexisting edges.

� The a1 term prohibits non-existent edges to be shared in thechosen path.

� The a2 term ensures that the inflow and outflow for each nodeexcept of the source and target nodes are equal.

� The a3 and a4 terms respectively, guarantee exiting of the sourcenode and entering in target node.

� Necessarily, the source and target nodes are always in solution.The edge from node d to s is a hypothetical one. The a5 term iszero when xd,s = 1. Since we exit from s and enter to d, thusthe final solution will always a loop which include s and d. Adirected path from s to d can be easily obtained from this loop.

� The a6 term does exactly in place of zero–one integrality con-straints to get a meaningful result.

3.2. Hopfield Implementation

Since in computer simulations, working on matrices with smalldimension is more convenience in comparison with that of largedimension matrices, we define a new variable vq instead of xi,j fori,j = 1,. . .,n. To join these indices, we utilize the following function:

f : f1; . . . ;ng � f1; . . . ; ng ! f1; . . . ;n2gf ði; jÞ ¼ nði� 1Þ þ j:

ð17Þ

As a simple mathematical exercise, one can show that f is onto andone-to-one function. Thus, this function can be utilized for our pur-pose, i.e. we set q = f(i,j) or equivalently vn(i�1)+j instead of xi,j in ourformulation of multi-objective shortest path problem. Similarly, thefixed parameters cp

i;j and ci,j are replaced with cpnði�1Þþj and cn(i�1)+j,

respectively. The original energy function (16) may be rewrittenas follows:

Table 1The first objective function (The elements of the matrix illustrate c1

i;j)

– – 0.1 0.8 – 0.5 – – – – – 1.0– 0.7 0.2 0.5 – 0.9 – 1.0 – – – 1.0– 0.2 0.2 – – – 0.4 – – 0.2 – –– 0.4 0.6 0.7 0.7 – – 0.3 – 0.4 0.0 –0.9 – – 0.8 0.3 0.8 0.5 0.9 – 0.8 0.8 0.7– 0.9 0.2 – – 0.7 0.4 0.7 0.4 0.7 1.0 –– 0.4 0.0 0.7 – 0.3 – – – – – 0.40.0 – 0.7 – – – 0.6 – – – – 0.90.8 0.1 0.4 0.8 – 0.3 – – 0.4 – 0.4 0.70.4 0.4 0.9 – – 0.5 1.0 0.2 0.2 – 0.5 –– 0.8 – 0.7 – 0.7 0.5 – 0.6 – 0.2 0.8– 0.0 – – 0.6 0.3 0.9 – – – 0.6 –

Table 2The second objective function (The elements of the matrix illustrate c2

i;j)

– – 0.7 0.1 – 0.4 – – – – – 1.0– 0.1 0.7 0.5 – 0.6 – 0.2 – – – 0.4– 0.0 0.1 – – – 0.9 – – 0.8 – –– 0.6 0.5 0.9 0.2 – – 0.9 – 0.9 0.3 –0.4 – – 0.3 0.8 0.5 0.1 0.5 – 0.8 0.7 0.7– 0.0 0.4 – – 0.9 0.8 0.8 0.6 0.4 0.4 –– 0.0 0.2 0.9 – 0.9 – – – – – 0.80.0 – 0.7 – – – 0.9 – – – – 0.50.0 0.6 0.7 0.8 – 0.2 – – 0.6 – 0.4 0.20.3 0.1 0.7 – – 0.9 0.7 0.4 0.7 – 0.4 –– 0.4 – 0.2 – 0.2 0.3 – 0.4 – 0.7 1.0– 0.6 – – 0.4 0.6 0.2 – – – 0.8 –

Fig. 4. A random network with three optimal paths keeping the first cost, thesecond cost and a combination of both of the costs.

4692 M. Ghatee, A. Mohades / Expert Systems with Applications 36 (2009) 4688–4695

EðvÞ ¼XP

p¼1

hp

Xn2

l¼1

cpl � vl � zp

�

!2

þ a1

Xn2

l¼1

cl � vl þ a2

�Xi–s;d

Xni

l¼nði�1Þþ1

vl �Xnðn�1Þþi

l¼i

vl

0@

1A

2

þ a3

Xns

l¼nðs�1Þþ1

vl � 1

0@

1A

2

þ a4

Xnðn�1Þþd

l¼d

vl � 1

!2

þ a5ð1� vnðd�1ÞþsÞ þ a6

Xn2

l¼1

vlð1� vlÞ; ð18Þ

To implement the Hopfield method and for obtaining necessaryparameters that are mentioned in (5) and (6) we apply the follow-ing signal function:

Sx ¼1 x P 0;0 x < 0:

�

We have:

oEovl�¼XP

p¼1

2hpcpl�

Xn2

l¼1

cpl vl � zp

�

!þ a1cl�

þ a2

Xi–s;d

2Sl��nði�1Þ�1Sni�l�Xni

l¼nði�1Þþ1

vl �Xnðn�1Þþi

l¼i

vl

0@

1A

� a2

Xi–s;d

2Sl��iSnðn�1Þþi�l�Xni

l¼nði�1Þþ1

vl �Xnðn�1Þþi

l¼i

vl

0@

1A

þ 2a3Sl��nðs�1Þ�1Sns�l�Xns

l¼nðs�1Þþ1

vl � 1

0@

1A

þ 2a4Sl��dSnðn�1Þþd�l�Xnðn�1Þþd

l¼d

vl � 1

!� a5Sl��nðd�1Þ�sSnðd�1Þþs�l�

þ a6ð1� 2vl� Þ: ð19Þ

Also

o2Eovlovl�

¼XP

p¼1

2hpcpl�c

pl þ a2

Xi–s;d

2Sl��nði�1Þ�1Sni�l�

� a2

Xi–s;d

2Sl��nði�1Þ�1Sni�l�Sl�iSn2�nþi�l

� a2

Xi–s;d

i¼1;...;n

2Sl��iSnðn�1Þþi�l�Sl�nði�1Þ�1Sni�l

þ a2

Xi–s;d

i¼1;...;n

2Sl��iSnðn�1Þþi�l�Sl�iSnðn�1Þþi�l

þ 2a3Sl��nðs�1Þ�1Sns�l�Sl�nðs�1Þ�1Sns�l

þ 2a4Sl��dSnðn�1Þþd�l�Sl�dSnðn�1Þþd�l � 2a6Sl�l�Sl��l: ð20Þ

Now we set

Wl;l� ¼ �o2E

ovlovl�: ð21Þ

Also for calculating Il we use Il ¼ � oEovlð0Þ instead of directly using

equation (6). We have:

Il ¼XP

p¼1

2hpcpl zp� � a1cl þ 2a3Sl�nðs�1Þ�1Sns�l

þ 2a4Sl�dSnðn�1Þþd�l þ a5Sl;nðd�1ÞþsSnðd�1Þþs;l � a6: ð22Þ

Example 3.1. Consider the edge costs of a random network inTables 1 and 2. The figure of this network is depicted in Fig. 4.

By using Dijkstra algorithm, the shortest paths with respect tothe first and the second edge costs are presented as follows:

1! 3! 7! 121! 6! 4! 12

associated with the 0.9492 and 0.8343 as objective function values.Keeping two objective functions and using the proposed Hopfieldalgorithm, the shortest path is

1! 12

with the cost 0.9908. In this case the weights of the first and thesecond objective functions are assumed to be 1/3 and 2/3

M. Ghatee, A. Mohades / Expert Systems with Applications 36 (2009) 4688–4695 4693

respectively. Also [a1,a2, a3,a4, a5,a6] are supposed as (Ahuactzin &Gupta, 1998; Balakrishnan et al., 2003).

Fig. 5. A configuration space with one obstacle.

Fig. 6. Creating a mesh. The clearance amount of each edge is associated with itselfas label.

4. Motion planning computation

An intelligent mobile robot in computer integrated manufactur-ing should be able to navigate without human control, performingfetch and carry tasks. Planning a collision-free path to each cell isone of the important requirements for robot to perform its tasks.Navigation itself is dependent on environment knowledge. Theability to build a map of an unknown environment is one of thefundamental enabling capabilities for mobile robots. The mapscan be presented as undirected graphs. In order to find a path tak-ing such graphs in mind, motion planing algorithms have been ex-tended. For instance, applying visibility graph, a breadth-firstsearch obtains a path by getting the source and target configura-tions. Such algorithms may be implemented in O(n2) applying asimple sweepline algorithm which can be improved to output-sen-sitive complexity O(k + n log n), where k is the number of edges incorresponding graph. Also, one can use reduced visibility graphtaking tangent edges into account with O(n + c2 log n) complexityin which c is number of obstacles, see e.g. (Sack & Urrutia, 1999).Also it is possible to find shortest path using Dijkstra algorithm.The same strategy can be used applying Voronoi diagram to finda path with maximum clearance. It is an interesting questionwhether or not there is a method to find a path with reasonableshortness and clearance for a mobile robot. It is trivial that, thesestrategies cannot be directly utilized for this aim. In order to findsuch paths, in Wein et al. (2007) a hybrid method taking visibilitygraph together with Voronoi diagram was introduced. But thismethod cannot implemented online.

To overcome on this difficulty, consider a roadmap with respectto obstacles, source node and target node. For simplicity a mappingalgorithm can be utilized to produce a grid where beliefs are en-coded, according to the sensor inputs. Each grid cell is assigned acertain value that it is free or occupied. In Lee & Chung (1994)for global path planning for an autonomous mobile robot in agrid-type world model such infrastructure was used. The value ofa grid cell representing the existence of an obstacle in the cell iscalculated from readings of sonar sensors. Once the world modelis obtained, a graph for path planning is built by using the model(see Lee & Chung, 1994 for details).

In the present paper, a uniform grid is used and both of the fol-lowing objectives are considered:

1. The length of edges,2. The farness from obstacles.

To define an appropriate grid, let h be the minimum distancebetween both of the given obstacles. For a given admissible toler-ance �, find a q such that h/q 6 �. Now consider a grid for configu-ration space including q as cell edge length. Such grid does notusually destroy the corridors of configuration space, but not al-ways. However, as � approaches zero, the accuracy increases.

Since a uniform grid is used, it is possible to take the length ofedges as unit. A cell is assumed to be occupied if it covers at leastone point of obstacle. Let O includes all of the occupied cells. Nowthe distance between each point u and each occupied cell O can bedefined as d{u,O} = min{d(u,v)jv 2 O}, where d(u,v) is the Euclideandistance between points u and v. Similarly the distance betweenedge (i,j) and O can be given by d{(i,j),O} = min{d{u,O}j u belongsto the segment corresponding to edge (i,j)}. If d{(i,j),O} = 0 we setd{(i,j),O} = �, where � is a given small positive value. In order tomeasure the clearance of edge (i,j), one can use the followingstatement,

ffði; jÞg ¼max1

dfði; jÞ;Og : O 2 O

� �;

It is clear that if edge (i,j) is barrier of an occupied cell, f{(i,j)} ap-proaches infinity.

Keeping these concepts into account, we should solve the fol-lowing 2-objective programming to find the optimal path for arobot.

minXði;jÞ2A

ffði; jÞgxi;j; ð23Þ

minXði;jÞ2A

ci;jxi;j; ð24Þ

subjected to the shortest path constraints.Thus, it is possible to use the proposed Hopfield algorithm to

solve this problem. Let a1 and a2 be the importance weights ofthe first and the second objective functions. Now, the applicationof presented scheme is shown for a sample example.

Consider the configuration space depicted in Fig. 5. The edgelength ci,j for each edge is assumed as unit. The details of f{(i,j)} cal-culation are represented in Fig. 6. The following two boundarycases may be taken into account:

Fig. 7. The result of most clearance path with no attention to shortness.

Fig. 8. The result of shortest path with no attention to clearance.

Fig. 9. The result of considering the clearance and shortness objective functionswith equal weights.

4694 M. Ghatee, A. Mohades / Expert Systems with Applications 36 (2009) 4688–4695

1. a1 = 1, a2 = 0,2. a1 = 0, a2 = 1,

It is important to note that, in the first setting all of the focus isdone on clearance of path, while in the second setting shortness isgotten attention. The result of implementing is depicted in Figs. 7and 8, respectively.

Now, assume a1 = a2 = 1/2. Taking the proposed scheme into ac-count we obtain the path depicted in Fig. 9. This illustrate the capa-bility of our scheme to provide different paths with respect to thedecision maker’s viewpoint.

5. Conclusion and future direction

In this paper we show how one can use Hopfield neural networkin order to solve multi-objective shortest path problem consideringTOPSIS scheme, as one kind of the goal programming techniques.Also we present a simple computational scheme for such imple-menting. The result of a random network is also illustrated whichclarifies the possibility of using this algebraic concepts in real envi-ronment. Then, by taking a uniform mesh into account, the motionplanning for a mobile robot is studied. Capability of online imple-mentation and taking the decision maker’s point of view, are twoimportant advantages of the presented approach. In the nextworks, we will study on implementing same methodology consid-ering other roadmaps into account. This approach can be directlyextended in unknown environment with time-varying conditions.Due to parallel implementation of neural networks, by applyingpath merging approach one can study on large-scale motion plan-ing problems in next researches.

Acknowledgement

We would like to express particular thanks to the honorableeditor for managing the reviewing process and also to the review-ers. The first author wish to thank Dr. M.R. Bank Tavakoli inDepartment of Electrical Engineering (AUT) for some comments.

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