356

THE DYNAMICS OF COLLECTIVE SORTING

ROBOT . LIKE ANTS AND ANT. LIKB ROBOTS

J.L. Deneubourgl, S. Gossl,, N. Franks2, A. Sendova-Franks2, C. Detrainl and L. Chrétienl

1 Unit of Behavioural Ecology , CP 231Université Libre de Bruxelles, 1050 Bruxelles

2 S.h*l of Biological SciencesUniversity of Bath, Bath BA2 7AY

Abstract

A distribured sorting algorithm, inspired by how

ant colonies sort their brood, is presenæd for use

by robot æams. The robots move randomly, do not

communicate, have no hierarchical organisation,have no global representation, can only perceive

objects just in front of them, but can distinguish

between objects of rrvo or more types with a cer-tain degree of error. The probability that they pick

up or put down an object is modulaæd as a func-

tion of how many of the same objeca they have

met in the.rêcent, pasL This generates a positive

feed-back that is sufficient to coordinate the roboe

activity, resulting in their soning the objecs inm

commolr clusters: While less eflicient than a hier-

archically controll€d sorting, this decenralised or-ganisation offers rhe advantages of simplicity,

flexibility and robushess.

l.Introduction

\Vhat is thecommon point bemeen a shopkeeper and an antcolony? Each of these organisms is able to sort similar butdiffsent objeets, When one examines an ant nest it is clearthat neither the workers, the brood nor the food' are ran-domly distributed. For example the eggs are arranged in apile next to a pile of larvae and a further pile of cocoons, orelse the threecaægories are placed in entirely different partsof ,the nesl The same is true in a shop. There is, however, anessential difference. The shopkeeper decides where he isgoing to put his different goods, and if he has assistants he

tells them where to place what. Ants work in parallel but donot, as far as we can æll, have the capacity to communicatelike the shopkeeper, nor do they have a hierarchical organi-sation whereby one individual makes the necessary deci-sions and the others follow. Nevertheless, if you tip thecontents of a nest out onto a surface, very rapidly the work-ers will gather the brood into a place of shelær and then sortitinodifferent piles as before.

This article describes a simple behavioural algo-rithm, to be followed by each worker, that generates a sort-ing process. Sorting is achieved without requiring eitherexternal heærogeneities (e.g. æmperatrne or humidity), hi-erarchical decision-making, cornmunication between the in-dividrrals or any global representation of tlte environmenlWe also stress that the antsÂoboB have only very local in-fstmation about the environment and a very short-termmemory, and furthermorc move randomly, no orientedmovement being necessary. They can't see far off,nor movedirectly towards objects or piles of objects.

Orn aim in this article is not !o prove that themodel proposed is actually how the ants,behave, blt toshow that such an algorittrm both works and could be usedby a team of robots. Inspired from our knowldege of theimportance of functional self-organisæion or disribuæd in-ælligence in ant colonies (pèneubourg, 1977; Deneubourget al., 1984, 1986, 1987; Deneubourg and Goss, 1989; Gosset al., 1990; Aron et al., 1990), our idea preserits a workingillustration of how such a distributed system can have prac-tical applications in robotics, in accordance with ideas de-veloped by ourselves (e.g. Deneubourg et al., 79M,lg0;Deneubourg and Goss, 1989), and others (e g. Beni, 1988;Brooks and Flynn, 1989; Sandini and Dario, 1989; Fukudaand Kawauchi, 1989; Brooks et al., 1990; Steels, 1990). The

Coilective Sorting: Robot-Like Ânts and Ant-Like Robots 351

environment for a real-time multi-robotic demonstration ofthis algorithm (and others related to ant-like foraging be-haviour) is under prepamtion at MIT, direcæd by MajaMataric and Rodney Brooks.

Firstly we describe a Monte-Carlo simulation ofthis collective behaviour, secondly ,fve present the continu-

ous mathematical model and its steady stâte analysis.

2. Monte Carlo Model

The model is based on the following principle. The ALRs orRLAs (ant-like robots or robot-like ants) move only ran-domly. When they come across an object the probability ofpicking it up is all the greater the more the object is isolaæd,i.e. the less the number of similar objects there are in theimmediate neighbourhood. When carrying the object anALR's probability of putting it down is all rhe grearer asthere are more of the same in the immediate neighbourhood.

Either of these two rules is sufficient to form separate clus-ters of the two object types, but both rules ûogether act muchfaster.

Isolated objecs are picked up. When a small, albeitloose cluster appears by chance, it "encourages" passingcarriers to add their load to the cluster thus increasing its"attractivity". With this positive feed-back mechanism theclusters grow, "absorbing" isolaæd objects and the smallerclusters through the action of the carriers. As clusters ofeach object type "attract" and fill up the nearby space withessentially the same type, this effectively isolates any objectof another type in the immediate vicinity, thus making itmore likely to be picked up by an ALR. Sorring is rhe con-sequence of this clustering and crowding out behaviour.

The environment is a square network of points. Attime zero a number of ALRs and objects of type A and Bare placed at random in the network, only one object and/orone ALR being allowed at each point. At each time step,and in random order, the AIRs move randomly north,south, east or west, alùough they cannot move into the wallaround the space or into a point already occupied by anALR. When an ALR moves onto a point containing an ob-ject it decides whether or not to pick it up. The less objectsof the same sort there are in the immediate enviroment, thegreater the probability rhat it will pick ir up, as given by thefollowing function:

where f is an estimation of the fraction of nearby points oc-cupied by objects of the same type, and k+ is a consrânt.The probability thus decreâses with f, from I (when f=0), tol/4 (when f= k+), and less as frends to l.

An anr could (probably) estimate f by the strengthof an odour associated with each bræd tlpe or else by tac-tile investigation, sight being less important æ such sortingis usually done underground. A robot would need a rathersophisticated visual, chemical or other sensory system to dothe same, and so we propose the following sampling-basedestimation which, while less precise, has the advantage ofbeing much simpler and more easily implemenæd.

Each ALR has a short-term memory of m steps,that records what it met in each of the last m time steps.Thus at t=10, a memory of length l0 could hold the string00AB0AA0B0, indicaring that during the previous ten rimesteps the robot mer 3 objects of type A and 2 of type B, theother points having been empty. f4 would be equal to 3/10,and fg to 2/10.

As the robot walks randomly, ttris sampling pro-vides a rough estimation of the density of the nllo sorts ofobjects in the immediate neighbourhood. Note that similarsampling techniques are known or suspected to exist in ants(e.g. Lumsden and Hôlldobler, 1983) and could possibly beat work in the way they decide to pick up or put down alarva.

As the memory is ten steps long, at the eleventhsæp the robot would forget what it met at the first sæp andadd what it met at the eleventh step. In this example tlrestring could become 0AB0AA0B0A if it next encountered at1rpe A object.

Whatever its decision, the ALR then carries on itsrandom walk. If it has picked up an object, then at each stepthat it finds itself in an unoccupied point it decides wheùeror not to put the object down. The more objecs of the samesort there are in the immediate environment, the greater theprobability that it will do so, as given by the followingfunction:

p (put down) = (f / ft- + fl;2

where f is as before, and k- is a constant. The probabilitythus increases wiù f, from 0 (when f=0), to lp $ = k), andmore as f tends to 1.

p (pick up) = (k+ / (k+ + f) )2

J. L. Deneuboure. ef a l .

Fig., 1,:Clustçring after f, 100000 and 2000000 sreps. 100 ALRs,1500 objects.,kf{..l, k:=0 3, m=50, ê0, space=J,90x200,points.Small evenly spaced clusters rapidly form, and laær merge intofewer larger clusters.

Frg. 2,,Chlstering :in: a colony ,of ;p11silefu pattiita:tà1.4000 corpseswere placed on a ,50x50cm aren4 and;,pft-etps,taken at,time,0;20and,68 hrs. Small evenly spaced clusters rapidly forral rrd,latermerge into fewer larger clusters.

i i r i ! iect i 'u 'E Scr i i rg : L+bca- i , ike Anrs and Ant-L ike I iobots

" i . i \ ' lo r r i t { l l r lo S imrr la l io l ls and CompârableI, ixgrcrinrents With Ànts

Fig. I slrows trow randomly distributed objects of one typearc mpidly gathcrecl into small and regularly.spaced clus-tcrs, which ovcr a longer period of time gradually mergernto a smallcr number of larger clusters. The clusters areconstztndy having elements removed and added, and there-lbrc tcnd to drift about slowly. When two clusters meet theyfusc.

Fig. 2 shows a similar process in a colony of phei-

dole pallidula. When ants die, workers carry the corpse outof the nest, and in laboratory conditions place them in a pile,a behaviour common to many ant species. In this experi-ment, a large number of ant corpses were spread out on anarena. Very quickly the workers (or robot-like ants) gath-ered them into a number of small clusters, which after along period of time merge inûo one or two large clusters(the experiment shown in fig. 2 did not run long enough forthis last stâge to be seen).

Fig. 3 shows how randomly distributed objects oftype A and B are rapidly sorted into small clusters of eachtype, which again over a longer period of time graduallymerge into a smaller number of larger clusters of each type.

The parameter values used have been selectedmore or less arbitrarily, helped by ttre fact that ttre modelsorts efficiently within a wide range of parameter values. Amore formal analysis of the influence of the parameûer val-ues will be performed elsewhere on the continuous modeldescribed below. It is nevertheless clear that a very longmemory length prevents effective sorting or clustering as itgives the ALRs the equivalent of a perceptive radius aslarge as the space they operate in, and so they could not dis-tinguish between an isolated object and one in a local clus-ter.

An interesting variant of the model introducessome overlapping, or imperfection of discrimination, be-tween the hilo sorts of objects, in the following sense. Anunloaded AI.R has met an object of type A and must tluscalculate f for A, or fg, being the number of objects A metin the last m steps divided by m. However one can introducesome confusion between A and B by calculating f4 as thenumber of objects of type A plus a fraction, e (the errorrate), of objects of type B, the sum being divided by m. inthe example above,00AB0AA0B0, f4 would become (3 +Ze)/tO. This is equivalent to the ALRs making a certain

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Fig. 3. Soning after 1, 60000 and 5700@ stçs. 20 ALRs, 200 oand 200 + objects, k+=0,1, k-=0.3, m=15,æ0, space80x49 points,Small evenly spaced clusters of each object rype rapidly form, andlater merge into fewer larger clusters with a high degree of sorting.

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Fig.4. Sorting afier 1, 225000 and 166@00 steps. 20 ALRs, 300 o

and 300 + objects, k+=0.1, k =0.3, m=15, e=0.2, space=80x49

points. Small evenly spaced clusters, containing both types of ob-

ject but placed adjacently, rapidly form, and later merge into fewer

larger clusærs, with a high degree of sorting.

Fig. 5. Sorting lh lMOm and72h after a aest of Leptothorax uni-

fasciatus has been mixed up. The lawae are rapidly gathered into a

number of small clusters, and later merge into one large clusær

with the small larvae (centre left) clearly separaæd from the large

larvae (centre and top and bottom left).

362 i. L. ûeneubourg, et al.

type A, they appear to have agreed to form a cluster of type

A at this point, whereas we know that no such conssnsus

exists.

By the same logic it is evident that one solitary

ALR can also form clusters and sort by interacting with its

own past actions, although it would act for example ten

times as slowly as a group of æn ALRs.

Another example of how such indirect"communication" can organise the activity of a group of

non-communicating agents can be seen in the way the for-

agers of some species of ants set up individual and non-

overlapping foraging territories (Deneubourg et al., i987).

Individual foragers learn progressively to retum to the area

where they find food. Vy'hen an individual becomes spe-

cialised to an area then by its activity it reduces the amount

of food there. Other individuals passing through that zone

will have less chance of finding food and so will be less en-

couraged ûo return to that zone by their own leaming pro-

cess. The same principle can also be used to allocate differ-

ent tâsks dynamically among the members of a group (see

also Theraulaz et a1., this volume).

Returning to the sorting context, it is clear that a

hierarchical system, wherein either a human supervisor or

an alpha-robot decides exactly where to put which type of

object, would sort more efficiently. However such a robot

would no longer be ant-like, and would require a capacity

for the analysis of how many types of object there are in the

environrnent and a means of communicating its decision to

other robots. All ttre robots would require at least a rudi-

mentâry map of the environment !o transport the objects

they find to the pre-arranged locations, or else would need

to home towards a beacon placed.bl some means at each lo-

cation. .Furthermore, any fluctuation in the environment

could make the original decision inappropriate.

The ALR system, while less effrcient, requires no

supervision and is capable of operating in a wide range of

environments without qpecific programming, and with a

large number of object types. The ALRs are only capable of

perceiving an object at the point they occupy, and have no

long range perception either for objecs, piles of objects or

homing beacons.

This simplicity makes the ALRs cheaper and more

robust" and the lack of any hieræchy prevents faal break-

downs, no one unit being essential. Even if a number of

ALRs broke down this would only slow and not prevent the

sorting. Again even if the ALRs frequently mistake one ob-

ject for another, sorting is unimpaired. Indeed, rather thansimply tolerating a certain degree of error, it can even bedesirable to deliberately add error where none or little exist.We have seen how a sma1l error in discriminating objectscan lead to the ALRs placing the pile or piles of the twosorts of objects adjacently. Another example in the socialinsect world shows how a certain degree of error in follow-ing nestmates' trails !o food sources allows ttre colony toexploit spatial heterogeneities in the food distriburion moreefficiently (Deneubourg et al., 1984). Put simply, if theyfollowed the trails too exactly they would never find newsources nearby the one they were guided to. Error can thusbe more creative than inefficient, and room should be al-lowed for it (see also the different works of Charles Dar-win).

Overall, the system's simplicity, flexibility, errortolerance and reliability largely compensate for their lower

efficiency. This is a general characteristic of systems inwhich the collective behaviour of a group of autonomousagents is emergent ratier than explicitly programmed, and issurely one of the reasons for the 100 million year long evo-lutionary triumph of social insects. There are many circum-stances in which a robot team could be profitably organisedin a similar fashion.

Acknowledgements

This wort was supported in part by the Belgian Program forInter-University Atnaction Poles, the Belgian Fonds Na-tional de Recherche Scientifique, NATO Scientific AfrairsDivision Grant 034/88, the Venture Research Unit of BritishPetroleum and the Ernst och Victor Hasselblads Foundation.V/e would like to thank Stefano Focardi for his helpful

scepticism.

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