Robot Motion Planning Utilizing Local Propagation of InformationBased on Particle Swarm and Its Internal Parameters

Gakuto Masuyama, Atsushi Yamashita, Hajime Asama

Abstract— A new artificial potential field method for motionplanning of mobile robot is developed in this paper. As areactive motion of robot, a feedback oriented to the contourof repulsive potential function is applied with the ordinalattractive and repulsive component. Furthermore, inspired byParticle Swarm Optimization, particles as simplified virtualrobots are utilized for motion planning. Each particle searchesthe space and transmit information regarding local stable regionto the swarm, namely, the robot and other particles. Internalparameters of the robot and particles are also introduced toadjust the propagation of information locally gained. Simulationresults demonstrate the robustness of proposed method againstthe complexity of an environment.

I. INTRODUCTION

Online robot motion planning is an important researchissue for autonomous robots operated in an unknown or un-predictable environment. In real time autonomous navigationproblem, robot must be capable of avoiding obstacles andreaching destination from initial state in a finite time interval.Enormous research has been developed and present robotnavigation methods would be classified into two categories(or their combination). That is, global path planning and localreactive motion generation [4].

In global methods, environment is assumed to be com-pletely known and the path is optimized in accordance withthe whole map information [10] [11]. Derived path can leadthe robot to the destination in a refined way. Furthermore,the path is collision and deadlock free, but they often requireenormous computational cost and do not consider dynamicsof robot. Therefore it might be ill-suited to adopt globalmethod and recompute optimized path for each variation ofa dynamic environment.

On the other hand, a whole environmental informationdoes not utilized in local methods [2] [8]. Although themotion of robot is not necessarily optimal, local methodscan be recomputed fast. In time varying and unpredictableenvironment, a capability to modify motion strategy rapidlyis one of the most fundamental specifications. The motionof robot should be determined by the trade-off betweendestination oriented stretegy and collision avoidance orientedstrategy. So a whole path efficiency depends the balance ofstrategies and therefore it is desirable for robot system to beable to recompute the motion at least in a comparable timeinterval with environmental change.

The authors are with the Department of Precision Engineering,Faculty of Engineering, The University of Tokyo, 7-3-1, Hongo,Bunkyo-ku, Tokyo, Japan. masuyama, yamashita,asama@robot.t.u-tokyo.ac.jp

However, local methods has potential problem to get stuckin trap situations like U-shape obstacles. It is difficult toguarantee the global asymptotic stability in principle becausethe motion is computed by limited environmental informationin limited time/spatial terrain. In this paper, we approach thisproblem through search actions of simplified virtual agents.Internal parameter of each agents entail social signal and themotion of robot is decided by sensory input and the spatiallypropagated information.

In the sequel, related works are discussed in Section II,and proposed method is presented in Section III. Simulationresults are shown in Section IV. Finally, discussion andconclusion are summarized in Section V.

II. RELATED WORKS

Proposed method is constructed based on artificial poten-tial field method. And ideas conceived by Particle SwarmOptimization is utilized to deal with some problems ofartificial potential field method. In this section, artificialpotential field method and Particle Swarm Optimization arebriefly reviewed with the objective of robot motion planning.

A. Artificial Potential Field Method

Artificial Potential Field method is one of the most pop-ular techniques for robot motion planning [2]. The fact isremarkable that this method can generate a smooth trajectoryand be easy to analyze mathematically. Additionally, precisemodel of an environment is not necessarily needed. Althoughartificial potential field method has these accessible proper-ties, there is local minimum problem of potential functions.A number of researches have been done to tackle with thisproblem, for recent example, [7] [9].

Let q ∈ Rn be ann-dimensional state of robot andUa :Rn → R+ andUr : Rn → R+ be attractive and repulsivepotential function respectively. Then the motion of robot isplanned by following equation.

q = −∇Ua(q)−∇Ur(q) (1)

If we choose attractive potential function as Lyapunov func-tion candidate, its time derivative is

d

dtUa(q) = −∥∇Ua∥2 −∇UT

a ∇Ur (2)

Thus (2) shows local equilibria can exists at such pointsqle ∈Rn that satisfies−∥∇Ua(qle)∥2 = ∇Ua(qle)

T∇Ur(qle). Solocal equilibria are always contained in the subspace ofRn,Ω = q : ∇UT

a ∇Ur ≤ 0, because−∥∇Ua∥2 is alwaysless or equal 0. Therefore two strategies can be effective toprevent trapping in local minima. One is evasion of approach

to the terrainΩ and the other is erasing local stable regionlike downward convex region.

B. Particle Swarm Optimization

Particle Swarm Optimization (PSO) [1] is one of a meta-heuristic optimization methods actively utilized for robot mo-tion planning in recent years [3] [5] [6]. PSO is inspired bysimplified social model of bird flocking and fish schooling.A set of randomly generated search points called particlesconstruct swarm and it propagates through the space. Eachparticle adjusts its motion in accordance with the actionhistory and information of search space shared across theparticles. Although PSO is mathematically simple algorithmand does not need information regarding gradient of objectivefunction, it can solve nonlinear optimization problem fast.

In this paper it is worthy of remark that the basic prin-ciple of original PSO is the hypothesis, “information isshared across the swarm”. Each particles utilize informationacquired not only by itself, but also other particles’ todetermine the search action. As a member of aggregation,it is reasonable to integrate the action pattern lead byown experience and the common sense for the purpose ofadaptation to the environment.

III. ARTIFICIAL POTENTIAL FIELD METHODUTILIZING PARTICLE SWARM

The robot is required to reach the goal state while avoid-ing collision with obstacles by real-time computation in anot fully-known environment. Particularly a local reactivemethod, artificial potential field method has suitable char-acteristics for this objective, but there is a local minimumproblem. Two strategies to prevent the trapped situation inlocal stable region would be possible as described in SectionII-A. In this section, proposed method is presented along theabove strategies.

A. Artificial Potentilal Field Method with Contour Feedback

Here, we discuss the evasion of terrain that possiblyincorporates the local stable equilibria. In artificial potentialfield, local minimum could exist in subspaceΩ = q :∇UT

a ∇Ur ≤ 0. ∇UTa ∇Ur ≤ 0 means the gradient of

attractive potential function and the gradient of summationof repulsive potential functions derived from each obsta-cle make an angle more thanπ. So roughly speaking, if∇UT

a ∇Ur ≤ 0 is satisfied, obstacles are frequently foundin the interspace between the robot and its destination. Ifrectilinear path is blocked by the obstacles, the robot mustgenerate circumvent motion in accordance with negativegradient ofUr. The success and failure of circumvent mo-tion depends on the existence of local minimum equilibria.Therefore it is desirable for the robot to set up not onlyrepulsive operator but also a explicit module to circumventthe obstacles that operates regardless of existence of localminimum equilibria.

Whereat we introduce a feedback oriented to the contourof repulsive potential function. LetJ ∈ Rn×n be a skew-

symmetric matrix, then the feedbacku is represented as

u = κ

(1− ∇UT

a ∇Ur

∥∇Ua∥∥∇Ur∥

)J∇Ur (3)

κ ∈ R is a parameter determines magnitude and directionof the circumvent motion. From (3),u could be regardedas nonlinear damper and it takes zero if∇Ua and∇Ur takesame direction. Henceu can generate motions that evade theterrainΩ. Besides, it makes no effect for the risk of collision,because it takes direction along the contour of repulsivepotential fucntion. With the feedback, whole dynamics ofplanner is modified toq = −∇Ua −∇Ur+u. Again, timederivative ofUa is

d

dtUa(q) = −∥∇Ua∥2 −∇UT

a ∇Ur

+κ

(1− ∇UT

a ∇Ur

∥∇Ua∥∥∇Ur∥

)∇UT

a J∇Ur

(4)

κ∇UTa J∇Ur can be positive if direction of circumvent

motion generated byu does not point a direction of thedestination. However, (4) shows it must be required to recedefrom destination in some trapped situations to move toglobal stable terrain. In the result it turns out sometimes therobot should tolerate detriment in short time interval for theaccession to the global minimum.

Introduced feedback works along the contour ofUr(q).It could be interpreted as a circumvent motion or searchmotion to discover terrainΩc = q : ∇UT

a ∇Ur > 0. InFig. 1(a), example of trajectory generated by the methodis shown. Square is a robot, cross is obstacle, five-pointedstar is destination and contour ofUa + Ur is also depicted.The robot go straight and encounter obstacles, then it startscircumventing to search other candidate of global stableterrain. Finally the robot succeeded to reach the destination.Fig. 1(b) shows the robot could be trapped by local minimumin enclosed situation by the obstacles. This is reasonablefunction because the robot should not muscle in to weaveits way through obstacles in such situations, especially whenobstacle represents human.

B. Particle Swarm as Virtual Agents

The robot sometimes generates an inefficient motion inabove method. As is shown in Fig. 1(a), if a wall standsbetween the robot and the destination, the robot would hustleagainst the repulsive potential field derived from the wall andfinally start to circumvent it. Obviously the derived path isnot the shortest one. This kind of problem is intrinsic forlocal reactive methods, because it makes decision withoutcalculating the entire optimality. However, as mentioned inSection I, global computation is not realistic in terms ofcalculation cost. Therefore a method enables the robot toobtain helpful local information efficiently is needed.

Information of the past best postion in the swarm is sharedacross the particles in PSO. At the same time, the particlesjust conform simple dyanamics but does not compute globaloptimality. Nevertheless PSO can be a powerful solver for

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Goal

Start

(a) Circumvent motion

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(b) Surrounded situation

Fig. 1. Artificial potential field method with contour feedback

an optimization problem. By metaphorical thinking of PSO,if the particles have dynamics as a virtual robot agent, itwould be able to get information concerning the terrain thathas the possibility to put the robot in undesirable situations.And if the particles can transmit the information of potentialrisk to the real robot in an appropriate way, the robot canutilize spatial information to decide its motion without globalcomputation. PSO has its basis on best action search likeforaging of living objects. In contrast, our idea has a basison worst action search like crisis prevention.

Let the dynamics of particles be determined by usualattractive and repulsive potential function as (1). Then theparticles would move toward the local or global minimumin accordance with their initial states. Therefore usage of justattractive and repulsive potential function provides spatiallylopsided information regarding equilibria in stable terrain.To utilize spacious and various information, we introducerepulsive potential functionU j

p : Rn × Rn → R+. j is anindex of m-tuple particles andqj

p ∈ Rn is a state ofjthparticle. ThenU j

p (qip, q

jp) is a repulsive potential function

assigned tojth particle and acts onith particle (whenq is

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Fig. 2. Contour of potential function compensated by three particles

chosen,U jp (q, q

jp) acts on the robot). A whole dynamics of

ith particle is represented as

qip = −∇Ua(q

ip)−∇Ur(q

ip)−

m∑j =i

∇U jp (q

ip, q

jp) (5)

So each particle is attracted by global minimum and repelledby obstacles and other particles. Thus information of thesearch space could be collected in reasonable manner, be-cause the particle moves as simplified real robot and repelleach other to extend the search space. Another merit to useparticles is that we have no need to consider the risk ofcollision, because they don’t have any physical body. So theparticles can search the space in a way real robot can notdo.

Next we design the propagation of information regardinglocal stable region to the real robot. This could be done bythe every assigned particles’ repulsive potential function thatworks on the robot. Particles search the space in accordancewith the dynamics (5) and might be trapped in local stableregion. This means unreasonable paths in terms of ordinalartificial potential field method exist in neighborhood ofthe particle. So the particle should offer the information tothe robot and other particles. That is, the particle generatespotential field that repells robot and other particles so as notto approach the terrain. Fig. 2 depicts particles, representedby upward-pointing triangles, move to the local stable regionand compensate it by assigned repulsive potential function.

With the effect of particles’ repulsive potential function,dynamics of robot is modified as below

q = −∇Ua(q)−∇Ur(q)−m∑j

∇U jp (q, qj) + u(q) (6)

u = κ

(1− ∇UT

a ∇Ur

∥∇Ua∥∥∇Ur∥

)J(∇Ur +

m∑j

∇U jp ) (7)

Hereat,u is replaced byu ∈ Rn represented as (7) thatalso circumvent repulsive potential functions assigned to

the particles. The reason of modification is thatU jp can

erase local minimum derived fromUr, but they also couldmake new local stable regions usually smaller ones than theoriginally existed. Note that the robot is not assigned anyrepulsive potential function act on the particles.

C. Internal Parameters and Proposed Method

Framework of proposed method has been constructed atthis point, but there could be some improvement from an-other point of view. As mentioned in Section III-B, proposedmethod has its basis on starategy of evading from worstsituation. Hence it might be possible to consider generatedmotion reflects events that operate the “emotions” of therobot and particles, so as to disincline the situation withina context of analogy with living objects. Here we label theinternal state as stress and discuss constructive utilizationof such function to make our method more adaptive to thestressful situations.

Firstly, we focus on dynamics of internal mechanisminside the particles. The task assigned to the particles isto inform the robot and other particles a prospect aboutlocal stable region. The information is transmitted by therepulsive potential functionU i

p centered on the position ofith particle. Whereat, width ofU i

p should be conditioned byhow undesirable the situationith particle is in. If there areno obstacles around the particle, thenU i

p should have narrowskirts so as not to block a path of robot and other particles.That is, ith particle does not feel risk or stress and thereare no information to share across the swarm. In contrast,if there are obstacles that block the path ofith particle, itwould be stressed by the obstacles. So the information ofstressful terrain should be informed across the swarm bywide skirted repulsive potential function. Here we setbip ∈ Ras a parameter of width of theU i

p. bip is adjusted by followingequation.

bip = βip tanh

(∫ t

t−Tp

exp(−λip∥qi

p∥)dt

)(8)

βip ∈ R andλi

p ∈ R+ are paremeters represent the maximumvalue of bip and speeds of stress accumulation.Tp ∈ R+ isa time interval stress is holded. So the stress accumulatedbeforet − Tp is got lost in oblivion. (8) meansith particlebroaden assigned repulsive potential function with it takessmall velocity. In local stable region a particle takes smallvelocity and takes zero at equilibrium, so the formulationis constructed asU i

p covers local stable region. Of course,U ip should not be expanded in neighborhood of global stable

equilibrium and it is treated special case. Actually, in Fig.2, stress parameters have been already intorduced and thetransition ofbip is depicted in Fig. 3.

Secondly, we discuss the design of internal mechanismfor real robot. Although the particles try to search the spaceand clarify risks of local stable region to the robot, theremay be situations in that the robot is trapped by local mini-mum of compensated potential functionUa +Ur +

∑mj U j

p .Alternatively, the robot might take cycled path and could

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t[s]

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p1

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p2

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Fig. 3. Internal parameters of particles

not approach the destination. Both of the cases degeneratesperformance of the system, but they could occur in unpre-dictable environment. To overcome such stalled situations,the robot should be capable of interpreting the situations andbreaking out from the terrains. And so we implement a stressparameters ∈ R+ and function to produce new particles tovary the neighborhood environment of the robot.

s =

∫ t

t−T

exp(λqT ∇Ua

∥∇Ua∥)dt (9)

λ ∈ R+ is a parameter regarding accumulation of stress andT ∈ R+ is time interval during that the past stress is holded.If the robot could not course steepest decendent directionof Ua during time intervalT and s exceeds the thresholdsth ∈ R+, then the robot lets out new particle, so the numberof particles is updated tom+1. The value ofs is initializedat the same time as new particle is produced. The sequenceis repeated until the particles release the robot from stressfulterrain.

Now then proposed method can be taken together by theequations (5), (6), (7), (8), (9). First of all, the robot placesome particles in the space and they move toward the des-tination along an attractive potential function. The particlesexpand skirts of assigned repulsive potential function if theyare blocked or stressed by the obstacles. The robot wouldproduce new particle if it could not proceed to the destinationcontinuously. In so doing, the motion of robot is generatedby the balance of attractive and repulsive and circumventingcomponents.

Note that there could be various design of stress functionsfor the robot and particles depending on the purpose ofrecusal. So (8) and (9) are only instances to exemplify theeffectiveness of proposed method.

IV. SIMULATION RESULTS

In this section, results of two computational experimentsare shown. One is an environment in that U-shaped obstacleis set. And the other is an environment in that multipleobstacles are set at random positions. Here, we utilize a

following potential function. The destination is set at origin.

Uk(q) = akqTq exp

(− (q − qk)

T (q − qk)

b2k

)(10)

ak ∈ R+ andbk ∈ R are parameters. When we takeq0 = 0and takeb0 large enough,U0 is treated asUa := U0. And(10) can be repulsive potential function if we takeqk as

qk =∥qi

o∥2 − b2i∥qi

o∥2qio

qio ∈ Rn is position ofith obstacle and thenUk has its local

maximum atqio. U j

p is represented in the same way forqjp.

Initial position of placed particle is determined as describedbelow.

q + rdqd − q

∥qd − q∥+ rufu

qd ∈ Rn is destination andfu ∈ Rn is a function outputsvector that has uniformly distributed elements in the interval(−1, 1). rd ∈ R+ and ru ∈ R+ are positive parameters.Particles basically precede the robot, because they shouldnot affect the robot motion immoderately and their functionis searching of potential field. In parallel, particles need tohelp the robot when it gets stuck in local stable region, so itwould be desirable to set the initial position of particle, orrd andru, at front of robot, not so far position.

A. U-shaped obstacles

To show the characteristic features of proposed method, anenvironment set U-shaped obstacle is simulated. Followingset of initial parameters are chosen.a0 = 0.5, b0 = 400,initial state of robotq0 = (0, 0), ajp = 0.5, bjp = 0.001,rd = 3, ru = 1.5, m = 4. Positions of obstacles are(6, 6), (5, 7), (7, 5), (4, 8), (8, 4), (3, 7), (7, 3), (2, 6), (6, 2),(1, 5), (5, 1), and corresponding(aio, b

io) are(1.5, 1), (1.4, 1),

(1.4, 1), (1.2, 1), (1.2, 1), (0.9, 1), (0.9, 1), (0.8, 1), (0.8, 1),(0.4, 1), (0.4, 1). T = 2, Tp = 2, J = [0 1;−1 0], sth = 1.8,λ = 1, βi

p = 1, λip = 0.1, κ = 0.5. Additional particles are

set byrd = 2, ru = 1. And the velocity of robot and particleshave their maximum at1 and2 respectively. Sampling rateis 100[ms] and whole simulation time is27[s].

Resulted trajectories of robot and particles are depictedin Fig. 4. Square represents robot, five-pointed star does thedestination, crosses do obstacles, upward-pointing trianglesdo initial particles, and downward-pointing triangles doadded particles respectively. Contour ofUa+Ur+

∑mj U j

p isalso depicted. In Fig. 4(a), robot and initial particles startedapproaching to the destination and go into interior of U-shaped obstacle. Obviously there exists local stable region,so bip ands increase gradually. Then the preceding particlescompensate the local stable region by increase ofbip, andthe robot starts circumvent motion without searching backof it. However, In Fig. 4(b), the robot moved few distances,because the local stable region could not be fully compen-sated just by the initial particles. So the robot is stressedand produce new particles in front at short time intervals.Finally in Fig. 4(c) the robot succeeded to circumvent U-shaped obstacle and arrived global stable terrain.

TABLE I

NUMBER OF SUCCEEDED TRIAL AND SUCCESS RATE

Original Contour ProposedCase 1 210 (70.0%) 267 (89.0%) 275 (91.7%)Case 2 67 (22.3%) 251 (83.7%) 279 (93.0%)

If we set more particles initially, generated path would bepre-circumvent motion and the performance should improve.Thus we can generate efficient motion of robot using appro-priate setting of particles. And even if improper settings isdone, the robot would produce particles in accordance withits stress parameter. This can be seen in Fig. 5 that depictsthe transition of internal parameters through the experiment.From the top of the figure,s and bip of two initial particlesandbip of three added particles are shown.

B. Randomly positioned obstacles

Obstacles are located in randomly determined positionsin following experiments. In particular, the initial robotposition and the destination are fixied at(0, 0) and (10, 10)respectively, and obstacles are uniformly distributed in anopen interval(1, 9) with respect to each axis. Number ofobstacle is also chosen as uniform random numbers from 1to 10 in Case 1 and 11 to 20 in Case 2. Number of trial is300 and each trial is executed at a time interval[0, 30]. Otherparameters are the same as the experiment of Section IV-A.If ∥qd − q∥ ≤ 1 is satisfied att = 30, the trial is judgedas success. Proposed method and original artificial potentialfield method and original method with contour feedback aretested. The number of succeeded trials and success rates aretabulated in Table I. It is remarkable proposed method havekept high success rate even though the complexity of theenvironment increased.

V. CONCLUSION

In this paper, artificial potential field method utilizingparticle swarm and its internal parameter has been presented.Particles embedded dynamics of ordinal artificial potentialfield method search local stable region. Each particle ac-cumulates stress as an internal parameter in local stableregion and it broaden assigned repulsive potential functionalong with increase of the internal parameter. Additionally,particles are added in accordance with the internal parameterof the robot to vary the stressful environment. The robot planits motion with the compensated artificial potential functionand a feedback oriented to contour of sum of obstacles’and particles’ repulsive potential functions. Computationalexperiments showed the effectiveness of proposed methodespecially in complex environment.

Narrow skirted and plentiful particles would generate pre-cise motion, but there is a trade-off between computationalcost and performance, so the evaluation of this term is futurework. And parameter of contour feedbackκ is fixed in thispaper.κ controls the direction and weight of circumventmotion. Obvioulsy performance of system could improveif it is selected appropriately. As a expanded virtual robot

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Fig. 4. Generated motion for U-shaped obstacle

body, or a kind of inverse model, another type of particlewould be available to determineκ. Finally, this method isoriginally invented as reference input for control systemof mobile robot. We are going to construct comprehensivesystem supposed for dynamic environment in the future.

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Fig. 5. Transition of internal parameters

VI. ACKNOWLEDGMENTS

This work was in part supported by The Ministry ofEducation, Culture, Sports, Science and Technology (MEXT)KAKENHI, Grant-in-Aid for Young Scientist (A), 22680017.

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