Surrounding robots –A localized solution for the intruder problem

Laszlo Blazovics 1 Tamas Lukovszki 2,1 Bertalan Forstner 1

1Department of Automation and Applied Informatics,Budapest University of Technology and Economics, Budapest, Hungary

Email: {laszlo.blazovics,bertalan.forstner}@aut.bme.hu2 Faculty of Informatics, Eotvos Lorand University, Budapest, Hungary

Email: lukovszki@inf.elte.hu

Abstract—In this paper we present a distributed localizedalgorithm for the problem, where a group of autonomous mobilerobots have to capture a target by forming a circle around it. Therobots do not have memory of the past (oblivious) and only uselocal sensing. Dedicated communication between the robots is notneeded. We introduce the discrete multi-orbit target surroundingproblem and present a solution for it. We prove that our solutionalways guarantees that the robots enclose the target and circulatearound it in O(D) time, where D is the sum of distances of therobots from the target in the start configuration. We also evaluateour solution by simulations.

Index Terms—mobile robots, synchronous model, target sur-rounding, localized algorithms, intruder problem, convergence

I. INTRODUCTION

Nowadays the field of cooperative mobile robotics has beenreceiving a lot of attention. There are several situations wherea group of relatively simple robots are more efficient than asingle intelligent one. Each entity make individual decisionsbased on their local information, where the overall groupconverges towards a common goal.

As one of the first researcher, Reynolds [1] modeled thecollective motion of a large number of self-propelled entities.He laid down the fundamental principles and achieved arealistic flocking model by the definition of the following threesimple rules, which should be individually obtained by eachvirtual bird: 1. Match the velocity of the other members of theswarm. 2. Avoid collision. 3. Avoid getting too far from theothers. The model was not only realistic but also required lesscomputational capacity than the former models.

In this work we introduce the discrete multi-orbit targetsurrounding problem, where autonomous robots with limitedsensing range have to intercept a target by surrounding it.The robots do not have memory of the past (oblivious). Wedescribe a localized algorithm to solve this problem, whereeach robot only needs information about the robots in its localenvironment and the target. We show the correctness and theconvergence of our solution. We prove that after O(D) stepseach robot reaches its final orbit, where D is the sum of initialdistances of the robots from the target. Our theoretical resultsalso hold without the need of communication between robots.We validate our theoretical results also by simulations.

This paper is organized as follows. Section 2 gives anoverview of related work. In Section 3 we introduce ourmulti-orbit surrounding algorithm and mathematical notations.We prove the convergence of the algorithm in Section 4and present an O(D) upper bound on the surrounding time.Section 5 presents our experimental results. Finally, Section 6summarizes the work.

II. RELATED WORK

There are many already working implementations of collec-tively behaving robot groups (a.k.a swarms) which relies onthe work of Reynolds like the RAVEN project of the MIT[3]the UAV system [4], or a military solution for UGVs [5].The main goal of these implementations is to design a stablegroup of quasi autonomous robots that can keep dedicatedformations.

From the perspective of the behavioral modeling one of themost simple solution is created by Mataric [7]. She defined fiveeasily adoptable basic rules. By using these five basic rulesmore dynamic and complex behaviors can be constructed.However the effectiveness of this solution was shown onlythrough simulations.

From the mathematical perspective there are many ap-proaches which are able to clearly formalize the cooperativebehavior. These theoretical results can be divided into twogroups: continuous time models and discrete time models.

Most continuous models are using artificial potential fields(generated by the environment and the entities itself) [8], [9],[10], [11], [12], in order to define interaction control forcesbetween neighboring robots and the environment itself. Thebenefit of this formalism was the easy provability of thestability of the overall system as it can be seen in the workof Gazi et al. [8]. Gazi have presented a virtual potentialfield based solution which are able to stabilize swarm ofhomogenous robots in an n-dimension Euclidean space. Basedon this approach Chu [9] has extended this to be able to handleheterogenous robots. However, by using interactive matrix, thefinal shape of the swarm will be rough in contrast with thesphere form of Gazi’s solution.

Leonard and Fiorelli [12] proposed a concept for the abovementioned problem. They introduced virtual leaders, that move

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CogInfoCom 2012 • 3rd IEEE International Conference on Cognitive Infocommunications • December 2-5, 2012, Kosice, Slovakia

”independently”, while they are followed by the rest of therobots.

Hsieh et al. [11] also used virtual potential fields in theirsolution in order to keep robots in a dedicated smooth shape”orbit”. Once the robots have reached the given orbit, they usethe tangential speed to avoid inter-agent collisions.

However the discrete time models are more theoretical,these models usually give upper bounds for the running timeof their behavioral algorithms the continuous models are onlyguarantee the convergence itself.

Cohen and Peleg [2] presented an asynchronous algorithmto gather robots at the center of gravity. Their algorithm usesthe LCM (Look-Compute-Move) discrete cycle based modelto move their robots. They mathematically proved upper andlower bounds on the convergence speed of their solution. Cord-Landwehr et al. [13] described an easy-to-check property oftarget functions that guarantee convergence and gives uppertime bounds. This property holds for the target functionin [2] and improves the upper bound on the speed of theconvergence.

Czyczowicz et al. [15] considered the gathering problem forfew fat robots, where the robots are modeled by unit disks. Thegoal was to gather the robots that the union of the unit disksis connected at the end. Collisions of the robots are prohibitedduring the gathering. A main problem which had to be solvedhere is that the line sight of a robot may be blocked by theextent of other robots. Cord-Landwehr et al. [16] studied theproblem of gathering mobile robots with an extent at a fixedposition as dense as possible to form a disk of minimum radiusaround the gathering point. The authors describe an algorithmfor the continuous case and the discrete case, where the robotmove on a grid. They prove a theoretic upper bound for theruntime of O(nR), where n is the number of robots and Ris the distance of the farthest robot from the gathering point.They empirically studied the continuous case, where in theyreport a few deadlock situations in the simulations. In oursurrounding problem the robots fill the inner orbits around thetarget and minimize the radius of the outermost orbit.

Chatzigiannakis et. al. [17] have presented a semi syn-chronous model by which a group of oblivious, individualsmall sized robots are able to form a circle around a givencenter point.

Gervasi et al. [18] have presented a solution for the intruderproblem. They have proposed an algorithm by which a groupof patrolling entities are able to detect and encircle an enemyunit. They use an asynchronous LCM-based model where alloblivious entities knew the position of the intruder and theother entities.

Prencipe et al. [6] give an excellent survey on theoreticalproblems and solutions in the area of cooperative robotics.

In field of sensor networks Li et al. [14] considered thesensor self-deployment problem for focused coverage, wheresensors have to cover an area around a fixed point withoutholes with maximal radius. They solved the problem ina discrete case on an equilateral triangle tessellation. Thisproblem is similar to our multi-orbit surrounding problem.

Fig. 1. Three circular orbits and their headings around a given target.

A key difference is that in sensor networks the sensors areable to communicate with each other within a limited range.Furthermore, during the sensor deployment the collision ofsensors was allowed, i.e. more than one sensors could occupythe same triangle vertex at at the same time. Li et al. [14]presented a proof of the convergence of their solution withinfinite time. The convergence time, energy consumption andnumber of collisions has been evaluated by simulations.

In this article we show a solution for the multi-orbit intrudersurrounding problem in the synchronous LCM model. Ouralgorithm rely on the position of the target and the neighboringentities only. We prove the convergence of our solution andgive an upper bound of O(D) steps on the running time, whereD is the sum of the initial distances of the robots from thetarget.

III. MULTI-ORBIT SURROUNDING

Surrounding is one of the most relevant tactic to capture amoving target, see e.g. [18]. Most animal groups hunting ingroup - like wolf packs - are following this strategy. Therefore,the main idea of our method is to define a orbit around thetarget on which the robots should take place.

In order to assure that the target cannot escape until therobots completely surround it, the surrounding robots shouldmove around the target. In order to avoid inter agent collisionswe have defined a heading direction on each orbit. As it isillustrated in Fig. 1., we have defined multiple orbits aroundthe target where neighboring orbits have opposite headingdirections. This accelerates the surrounding process, i.e. if aninner orbit contains a hole, an additional robot from the nextorbit can fill it in shortest time.

A detailed description about our basic concept can be foundin [19].

The key difference between the basic concept in [19] andthis fully discrete model is the deployment and the movementof the entities. In this model the orbits are regular polygons(see Fig. 2.), in contrast with the basic circle form. The ithorbit is a regular polygon of 8i vertices, i = 1, 2..., with sidelength of one unit, such that the center of this polygon iscoincident with the target. The robots can occupy the verticesof the polygons. In each time step, a robot can move to aneighboring vertex of the same orbit or to the closest vertexof the next polygon in the direction of the target. With other

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Fig. 2. The alignment of the nodes on the first two trajectories.

words we define a directed planar graph, in which a robotcan traverse an edge in a time step. We call this graphpolygon graph. The vertices of this graph are the vertices ofthe polygons. A vertex of a polygon has a directed edge tothe neighboring vertex of the same polygon in the headingdirection assigned to the polygon (solid line segments in Fig.2) and to the closest vertex of the next polygon in the directionof the target (dashed line segments in Fig. 2). If a vertex isequally far from two vertices in the next polygon (like n2,2

from n1,1 and n1,2) we have an edge to the vertex on the theleft-hand side.

The polygons are numbered incresingly started by one withincreasing distance from the target. The vertices of eachpolygon Ti are numbered increasingly in clockwise orderstarted by one. The vertices of polygon Ti are denoted byni,1, ni,2, etc...

The goal of the entities is to place themselves on theinnermost orbit which has at least one free node. Similar tothe basic concept, if an entity is unable to the get closer tothe target – because of another entity is in front of it, or it hasreached the innermost orbit – it should circulate on the currentorbit by jumping to the neighboring node. In order to preventthe inter-agent collisions during the circulation, each orbit hasa predefined heading direction. If the entity is able to moveto the next inner orbit it should check whether an other entitytries to get to the same vertex – like the entities on n2,2 andn2,3 in Fig. 2. can move to n1,2. More precisely, followingmust be checked. A vertex ni,j can be occupied in the nextstep from robots on the vertices having a directed edge to ni,j .There are at most three such vertices, one in the same orbitTi and at most two on Ti+1. Thus, at most three robots canstay in conflict. In case of a conflict, the robot on the vertexwith lexicographically lowest index can occupy ni,j and theother(s) must circulate. Thus, each robot can check locally,whether it can move towards the target or it must circulate ifthe corresponding vertices are within its sensing range.

These rules imply that that the nodes either move towardsthe target or circulate around it, however they never stay onthe same vertex longer than one time step.

IV. ANALYSIS OF THE CONVERGENCE

In this section we prove that by using of the above in-troduced surrounding method a swarm of robots will alwaysenclose a given target. For this we use following assumptions.

A. Assumptions

1) Given n robots, modeled as points in the Euclidean planeIR2, taking place at the vertices of the polygon graph,wich is described in the previous section.

2) The robots use a common coordinate system.3) Each robot knows its own position and the position of

the target t. W.l.o.g. the target is placed at the origin.4) The movements of the robots are divided into discrete

steps.5) In each step a robot traverses a directed edge of the

polygon graph.6) The sensing range of the robots on polygon Ti contains

the neighboring vertices in Ti, the closest vertex in Ti−1,if i > 1, and the vertices having a directed edge to thisclosest vertex in the polygon graph.

Remark: For simplicity, we assume a static target. Thisassumption can be changed for a slowly moving target. In thatcase the vertices of the polygon graph move with the target.

B. Algorithm description

We assume that at the beginning each robot resides ondifferent nodes and tries to move on the edges towards ofthe target.

We say that two robots u and v are in conflict and maycause collision if they can move to the same vertex. In thatcase the robot on a vertex with lexicographically lowest indexis allowed to move to this vertex. The algorithm is presentedbelow.

Algorithm 1 Multi-Orbit Surroundingloop

if current position is on T1 or there is a robot ona vertex with lexicographically lower index having thesame destination then

Circulate on the orbit around the target in the corre-sponding direction

elseMove towards the target

end ifend loop

We prove that the robots always can move, never stuck indeadlock situation and we give a convergence guarantee of thesurrounding process.

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C. Convergence

First we show that each robot can move either into thedirection of the target or on the orbit around the target.Therefore, the sum of the distances between the nodes andthe target never increases during the process.

Lemma 4.1: Each robot v, which is not on the innermostorbit, can move towards the target if v is not in conflict withanother robot u with higher priority. Otherwise, v can moveon the orbit around the target in the corresponding headingdirection. The distance between the target and the robot neverincreases during the surrounding process.

Proof: First we consider a robot v which is already on theinnermost orbit T1. We show that v can move on that orbit andits disctance never increases. Since all robots on T1 (if any)move in the same direction around the target, their distanceto each other remains the same, and thus, they do not causea collision. Another robot u is only allowed to move to T1, ifit does not cause a collision.

Now we consider a robot v which is not in T1. If v is notin conflict with any another robot u then it moves towards thetarget and its distance strictly decreases. Otherwise, by similarargument than above, v can move on the orbit correspondingto its current position and its distance to the target does notchange.

Now we are able to prove a guarantee of the convergenceof the surrounding process.

Theorem 4.2: Until the vertices of the inner orbits have notbeen completely occupied by robots (i.e. an inner orbit Tin

contains an unoccupied vertex, the sum of distances of therobots from the target strictly decreases by at least one within(i+1)· 8+1 steps, where i is the index of the of the outermostorbit Ti.

Proof: Our rules guarantee that for each robot the distancefrom the target never increases. If a robot is not prohibited byother robots, it moves towards the target, until it reaches theinnermost orbit. If a robot does not decreases its distance tothe target in a time step, then it is either on the innermost orbitor it is in conflict with another robot.

Assume that in a time step no robot can decrease its distancefrom the target. Let Ti be the innermost orbit which containsan unoccupied vertex. Consider the robot v further fom thetarget than the vertices of Ti, which is on the vertex withlexicographically the lowest index. If no such robot exists, thenwe are done, all inner orbits are fully occupied. Otherwise, vcan only be prohibited to move to the direction of the targetby robots on Ti. Then v decreases its distance until it reachesorbit Ti+1 where it starts the circulation. The robot v and theunoccupied vertex on Ti circulate in opposite direction. Withini· 8/2+1 steps either v can move to the unoccupied vertex inTi or another robot occupied it before v. Thus, within i· 8/2+1steps at least one robot strictly decreased its distance to thetarget.

Theorem 4.3: After O(D) time steps all inner orbits arefilled, where D is the sum of distances of the robots to thetarget in the starting configuration.

Proof: For i ≥ 1 and j ≥ 1, let ti,j be the time, when allorbits T`, ` < i, are already fully occupied and the numberof robots on orbit Ti increases from j − 1 to j. The robotvi,j joining the orbit Ti at time ti,j remains on Ti, since allorbits closer to the target are already completely filled. Fori ≥ 1, let mi be the number of robots on Ti, when all orbitsT`, ` ≤ i, are completely filled. Let Vi,j be the set of robotsthat have already reached their final orbits til time ti,j . Tosimplify the description let ti+1,0 := ti,mi , i ≥ 1 and t1,0 bethe starting time. Let dti,j := ti,j− ti,j−1, i ≥ 1, 1 ≤ j ≤ mi.Let ui,j ∈ V \Vi,j−1 be the robot at time ti,j−1 which wouldnot be in conflict with V \ Vi,j−1. The robot ui,j can movein the direction of the target until it reaches the orbit Ti+1 ineach time step. Then at most after O(i) steps of circulationon orbit Ti+1 it can occupy an unoccupied vertex in Ti, if Ti

has not been occupied before this time step. Thus, we obtainthat dti,j ≤ di,j−1(ui,j , o) + O(i), where di,j−1(ui,j , o) isthe distance of ui,j and the target at time ti,j−1, which isnot greater than the initial distance d1,0(ui,j , o) of ui,j andthe target. Furthermore, d1,0(ui,j , o) of ui,j ≥ i. Therefore,dti,j ≤ c · d1,0(ui,j , o), for some constant c > 1. Thus we canconclude that the time t until all the robots reach their finalorbit is:

t =∑i

∑j

dti,j ≤∑u∈V

c · d(u, o) = O(D).

V. VALIDATION AND SIMULATIONS

We have performed simulations for various robot swarmsizes and startup constellations. The target was placed at thecenter of a square simulation area. We have used three differenttype of start-up constellations. The first one is the randomconstellation, where the robots are distributed uniformly atrandom in the simulation area. The second setup is the twogroups constellation, in which the robots form two separatedense clusters.The third constellation is the one group setup.In this case the robots are placed in one dense clusters. Inthe clustered settings many robots were in conflict with eachother at the beginning. In all cases we have made differentmeasurements regarding the number of the entities and thesize of the simulation area. We have used 20-100 robots andthe side length of the simulation area was 20-40 unit.

Figure 3. shows the simulation results. The surroundingtime increases when the robots concentrate at a specific sideof the target, like in the one group setting. In the randomcase, surrounding time manly depends on the average distancebetween the target and the robots. The peaks on Figure 3(a)are caused by the disadvantageous position of those robots thatarrive last to the target. In a worst case situation a robot mustmove around the half of the perimeter of an inner circle untilit can move into a hole on it.

We have also performed simulations in which all robotsform a circle around the target in an outer orbit with 1 unitinter-agent distances. Despite the conflicts between the robotsat the beginning, the robots have reached the final constellation

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(a) random (b) one group (c) two groups (d) circle

Fig. 3. Surrounding time for the different type of start-up constellations. Number of robots: 20-100, side length r of the simulation area: 20, 30, 40.

within a time linear in the radius of the starting circle as it isshown in Figure 3(d).

The simulation results suggest that the robots reach the finalconstellation even faster than the proved theoretical bound.The reason is that our proof does not exploit the fact thatwithin a time step many robots can decrease the distance tothe target simultaneously.

VI. CONCLUSION

We have introduced the discrete multi-orbit surroundingproblem, where autonomous robots with limited sensing rangehave to surround a target and circulate around it. The robotsdo not have memory. Communication between the robotsis not needed. We have described a synchronous algorithmto solve this problem. We have proved that our algorithmalways guarantees that the robots enclose the target andcirculate around them in O(D) time step, where D is thesum of distances of the robots from the target in the startconfiguration. The simulation results suggest the robots reachthe final constellation even faster than the proved theoreticalbound.

ACKNOWLEDGMENT

This work is connected to the scientific program of the”Development of quality-oriented and cooperative R+D+Istrategy and functional model at BUTE” project. Thisproject is supported by the New Hungary DevelopmentPlan (Project IDs: TAMOP-4.2.1/B-09/1/KMR-2010-0002,TAMOP-4.2.1/B-09/1/KMR-2010-0003).

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