2008 10th Intl. Conf. on Control, Automation, Robotics and VisionHanoi, Vietnam, 17–20 December 2008

Trophallaxis in Robotic Swarms - beyond EnergyAutonomy

(Invited Paper)

Henrik SchiolerCenter for Embedded Software Systems,

Section for Automation and ControlAalborg UniversityAalborg, Denmark

henrik@control.aau.dk

Trung Dung NgoCenter for Embedded Software Systems

Aalborg UniversityAalborg, Denmarkdungnt@cs.aau.dk

Abstract—The paper considers trophallaxis in robot swarms,which is presented as a concept for resource sharing amongindividuals inspired by altruistic behaviour in natural popula-tions. A number of elementary problems are identified, suchas energy containment, robot morphology, rendezvous motioncontrol and individual resource exchange behaviour. The CISSBotis presented as a design study as well as a proof of conceptillustrating how a trophallactic exchange mechanism may beimplemented based on batteries in commercially available formfactor. A collison free proximity motion control is presentedand illustrated by numerical simulation results. A probabilisticMarkovian model including relevant effects; mobility, charging,battery exchange and energy consumption is presented andillustrated with numerical examples.

Index Terms—

I. INTRODUCTION

Collaboration between individuals in groups is a frequentlyappearing phenomena in nature as well as among humans. Itserves a variety of tasks for the benefit of the group as a wholeas well as for the individuals; worker ants participate in theconstruction and maintenance of the nest as well the feeding oflarvae and the queen, mongooses collaborate in babysitting andsurveillance and snakes group in pits during dormant periodsto reduce heat loss. A very basic kind of collaboration isconstituted by the gathering and distribution of resources suchas food among members of the group; lions collaborate duringhunting and share prey subsequently, altruistic behaviour isfound among birds where individuals of higher status feedlower status individuals. The origin of collaborative/altruisticbehaviour is still an open scientific subject. From a gametheoretic perspective individuals may profit from an altruisticreputation [3], i.e. it is probably worthwhile sharing withsuch individuals or at a higher level groups of predominantlyaltruistic individuals may exhibit a higher survival rate and aretherefore favoured in an evolutionary process. Others suggestaltruistic behaviour as an element in a strategy of advertisingstrength to improve reproduction probability [2], which in turnis favoured from an evolutionary perspective.At a general level most studies in robotics is occupied withthe transfer of human skills and behaviour to the robotic coun-terpart; industrial robots replace human work force and newly

developed social robots with imitated mimics substitute humancompany for elders or mentally disabled. In this perspective thetransfer of altruistic behaviour to groups/swarms of mobilerobots seems a natural extension to this effort, which is notonly motivated as a study object but indeed as a means ofimproving efficiency and increasing autonomy - not of theindividual robots - but of the swarm as a whole.It is a well known fact that energy resources constitute akey bottleneck on the way towards efficient deployment ofmobile robots in real life enviroments. Most mobile robots arebattery powered and equipped with a rechargeable battery andfrequently in the need of recharge at one among a small num-ber of charging stations. Charging stations may them selvesconstitute a bottleneck, i.e. it may prove impractical/expensiveto install a large number of such stations, in which case theymay be separated by a great distance from each other aswell as from robots. Since travelling to charging stations itself consumes energy as well as time, such a situation mayprove intolerable. Also a small number of chargers serving alarge number of robots may produce a bottleneck situation,where robots have to sustain waiting at the charger due tohigh charging times. Such disadvantages may be reduced byallowing altruistic/trophallactic behaviour among robots. Weenvision a small number of charging stations serving only therobots in their proximity. Recharged robots in turn exhangeenergy resources with robots in their own proximity. Thiscreates a systematic flow of energy resources from chargingstations to the distant parts of the working domain, allowingrobots, assigned to work tasks in such remote areas, to maintainwork without frequently having to resort to charging.To facilitate energy sharing among robots several technicalproblems need to be overcome; energy needs to be containedin a form which is exchangeable between robots, movementcontrol is required to facilitate rendezvous between robots aswell as between robots and chargers, this in turn requiresmeans for global and local navigation, the morphology ofrobots should allow the physical exchange of energy containersand finally an exchange policy/behaviour should be designedto optimize predefined efficiency objectives for the group asa whole. This paper is designated to the discussion of these

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topics and the remainder is organized as follows: the followingsection presents and discusses a design example of a physicalplatform for robot trophollaxis - the CISSBot, this is followeda related work section and a section devoted to the design of arendezvous motion controller based on linearization of a localrelative coordinates. Hereafter a section is devoted to a prob-abilistic model of trophallaxis including, charging, exchange,mobility and energy consumption. Finally conclusive remarksand summary is given along with suggestions for the directionof future research.

II. CISSBOT

As a part of our research in trophallactic robots, we arecurrently developing a prototype of a robot facilitating exhangeof energy resources, called CISSbot shown in figure (1). TheCISSbot is physically formed in a wXl rectangular shape,where length l = 15cm and width w = 15cm. It is adifferential two wheeled robot with a special mechanism ofbattery exchange. The architecture of the CISSbot is dividedinto two main parts: the lower layer and the upper layer. Thelower layer is the behavioral control to perform the behavioralautonomy while the upper layer supports battery exchangeto refuel the robot’s energy or to share energy with others.The lower layer of the CISSbot also has a full specificationof sensors, actuators and a central processor to control therobot’s behavior automatically. However, unlike mobile robotsseen previously, we have been developing a battery exchangemechanism for the CISSbot to load and unload batteries.The mechanism is designed in an H-shaped set of batteryholding boxes that include the electrical contacts and thebattery pushing systems at both sides.

Fig. 1. CISSBot bird perspective view.

A. Energy exchange

The mechanism of battery exchange is a complicated ar-chitecture. To protect against short-circuits, flexible plastic isthe material chosen to compose the architecture. In particular,the skeleton of the battery exchange mechanism is assembledfrom several flexible plastic parts to form 8 battery boxes onthe base. At the two sides of each box, there are two brassplates to physically conduct electricity from the batteries to theentire system. To hold a battery in the battery box, as well as toprotect against mechanical shocks when the robot is moving,a plastic pad is tilted 12.5 deg over the central base (1). Eachbattery box involves a linear motion mechanism fixed at the

two ends of the box in order to allow the mechanism to pushthe battery in the slide-way.

Fig. 2. CISSBot exchange mechanism.

B. Power management

To facilitate a qualified energy resource exchange theresource level of robots engaging in exchange constitutesindispensable aspect of information. The main tasks servedby the power management system are to continuously sensethe current state of the batteries carried by the robots, protectagainst occurrence of the short-circuits, regulate the power ofthe various sources of usable energy for the robot’s actions,e.g. select the number of batteries in use.The current state of the battery is of major importance forsmart energy management. Depending on the chemical char-acteristics of the rechargeable batteries, e.g., Li-Po, Li-Ion,Ni-CD, Ni-MH, etc., their capacity is changed differently,and thus their discharging functions are not identical. Thismakes battery monitoring more difficult and less precise ifonly the voltage of the battery is measured, which is whatmany battery monitors have done previously. Moreover, loadof the system powered by the battery probably affects themeasurements. The question is, how to monitor the currentstate of different types of battery precisely. To improve theefficiency of energy management we have chosen to monitorboth the current and the voltage of the battery. Fortunately, thismethod has also been the considered by Maxim electronicsoffering a commercially available smart battery monitor chip.We need to connect only one wire of the chip to our processorin order to evenly monitor the elements: voltage, current, andtemperature.

C. Navigation

To guide robot motion in a meaningful way some meansof navigation is surely required. Global navigation is neededto serve location dependent tasks as well as for locatingcharging stations when needed. Several solutions to theglobal navigation problem have been proposed based onoptics, radio communication as well as ultra sound. A finaldecision for the global navigation for the CISSBot is still anopen question. Local navigation is required when robots byrandomness come into mutual proximity and it is decided toengage in resource exchange. As described in the followingsection only local relative coordinates are required in thisphase, which is accomplished by the inclusion of a dual axismagnetic field sensor to act as a electronic compass. External

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magnetic anomalies are assumed to be of less significanceto measurement accuracy when only relative orientations areneeded. For relative position the CISSBot is at each sideequipped with four pairs of infrared proximity sensors. Thecurrent design allows sensing up to 16 cm in front and onboth two sides of the CISSbot approximately. Additionallya CISSbot is also set up with incremental encoders forboth wheels. The encoder will increasingly accumulate thetravelling distance by observing a barcode, which is fixedon the wheels, when the wheels are rotating. In the currentdesign, the sensor is able to sense 2 mm a long the distanceof the robot’s movement. Since the encoders are incremental,positional errors integrate along time. However they maybe kept below satisfactory levels within the duration of oneinstance of the resource exchange proces.

D. Communication

The CISSBot is equipped with communications facilitiesfor both close and short range communication. Local closerange communication is achieved through infrared photodi-odes located at the side of the CISSBot. These are intendedfor direct communication with the partner in resource ex-change rendezvous, which relieves the global radio frequencycommunication channel of unessesary communication. Globalcommunication across the entire work domain is achieved byadding a 2.4 GHz radio module operating in the license freeISM band. Even though RF communication is short rangewe envision ad hoc networking among robots to facilitatecommunication across a domain of operation extending therange of a single RF link.Proximity to other robots candidating for resource exchangeis detectable through communication facilities; namely whena robot is within range it is considered to be within proximity.For RF communication received signal strength may indicateproximity.

III. RELATED WORK

In this section, relevant related work within behavioral andenergetic autonomy is addressed. We have chosen examplesbased on similarity to our design specifications. For behavioralautonomy, it is easy to search for a list in terms of self-reconfigurable robots and behaviorally autonomous robots. Theformer is defined as a machine built from several identicalmodules, and by reforming their module connections to changethe shape and functionality of the entire machines or organism.The M-TRAN [11],[12] and ATRON [13], [14] are reconfig-urable robots and typical representatives of the class. M-TRANmodules are battery-powered units but they are not able toshare or self-recharge energy automatically, while the ATRONmodules are also battery-powered units that are able to shareenergy by inter-connections. Swarm-bots (s-bots) is a Europeanproject dedicated to the creation of self-assembling and self-organizing robots inspired by social insects or other animalsocieties. Unlike M-TRAN or ATRON, s-bot is a completelyindependent mobile robot, that is, s-bot is able to move without

the need to inter-connect with other s-bots. For capabilities ofself-assembling and self-organizing, s-bots are equipped witha flexible gripper that is capable of grasping another s-bot tolift it when it needs to pass over large holes or pass throughnarrow passages in complex environments. They also have twoshort rigid grippers to provide easy connectivity to the otherrobots. In combination with two kinds of gripper, several typesof sensors are used to play very different roles in swarm-robot configurations, e.g., infrared proximity sensors, infraredground proximity sensors, color sensors, inclinometer sensors,humidity and temperature sensors, etc. Swarm-bot has no capa-bility for sharing energy among robots, or refueling its energyby exchanging batteries with a charging station, eventuallythey would be able to perform such capabilities thanks to veryhigh processing power, excellent sensing capability, and smartgrippers. In short M-TRAN, ATRON, and Swarm-bots are notyet truly autonomous robots. However, features of physicalconnections, neighbour communications, and processing powerare very useful examples when designing the CISSbots. Forenergetic autonomy, there exist only a few robots that areable to act as energy rechargeable robots or self-poweredrobots. A good example of the first class are vacuum cleaningrobots like Roomba and CleanMate. Typically, the robots movearound freely to clean carpets and automatically return to thedocking station to recharge the battery when it is low. Althoughthe robots have partly solved the problem of self-rechargingenergy, they still lack the capability of quickly recharging byexchanging batteries instead of waiting at the charger for a longtime. Also, the robots can only work alone without capabilityof sharing energy among moving robots as our CISSbots do. Awell-known robot of the second class is a series of ecologicalrobots named EcoBot. These robots are referred to as a classof energetically autonomous robots that are self-sustainable bycollecting their energy from waste in the environment. EcoBot-I [15] was developed to utilize sugar as the fuel by onboardMicrobial Fuel Cells (MFC), while EcoBot-II [16] was createdto consume dead flies, rotten fruit, and crustancean shells. Theapplication domain of EcoBot is places which are very hard toaccess by humans, e.g., under water or in a poisoned area. Alsothe EcoBot has partly overcome the problem of self-powering,but they can only work in a waste environment where biomassis available. Moreover, like the vacuum clearing robots, theEcoBot usually acts alone, so it can not be able to benefitfrom colony activities.

IV. MOTION CONTROL FOR RENDEZVOUS

Energy exchange is in this work considered to be anunplanned epidemic process, i.e. transfer of energy betweenrobots take place during accidental rendezvous. Epidemicpropagation is previously studied in other contexts, such asdisease spread [9] and information spread [8],[10]. Thereforewe may assume robots to be within mutual proximity, whenresource exchange is decided, which is taken into accountduring design of the motion control.Since the CISSBot is driven by a differential two wheeledsystem our mechanical modelling and motion control design

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is aimed at this mechanical design.

A. Kinematic Model

The kinematic model is based on the definition of a positionvector x and a unit direction vector q, which constitute the stateand outputs of the dynamical system. Inputs are constitutedby angular velocities wright and wleft of right and left wheelrespectively or more directly wm, ws defined by

ws =r

2(wright + wleft)

wm =r

2d(wright − wleft) (1)

where r denotes wheel diameter. From the definition of in-puts a kinematic model expressed in cartesion coordinates isobtained, i.e.

x = q ws

q = R q wm (2)

where R denotes a 90 degrees counter clockwise rotationmatrix.

B. Trajectory Tracking

For (2) various nonlinear control designs may be suggested[19],[20],[21],[22] However, if a smooth reference trajectoryis defined, a local linearization is possible and facilitates thedesign of linear control laws. For a smooth reference trajectorywe define reference inputs and states, i.e. wr

s(t), wrm(t) and

qr(t), i.e.

wrs(t) = |xr(t)|

qr(t) = xr(t)/|xr(t)|qr(t) =

xr(t)|xr(t)| − qr(t) < xr(t), xr(t) >

|xr(t)|2wr

m(t) =< qr(t), R qr(t) >

which solve the feed forward trajectory following problem.Computing differences

x − xr = q ws − qrwrs

q − qr = R (q wm − qr wrm)

and linearizing, gives

x − xr ≈ (q − qr) wrs + qr (ws − wr

s)q − qr ≈ R (q − qr) wr

m + R qr (wm − wrm)

Finally, by transformation to local coordinates

xa =< x − xr, qr >

xp =< x − xr, R qr >

qa =< q − qr, qr >

qp =< q − qr, R qr >

we obtain

xa = qa wrs + ws + xp wr

m

xp = qp wrs − xa wr

m

qa = 0qp = wm

, i.e. a linear time variant dynamical system. For wrs > 0 we

may choose

ws = −Ka xa − qa wrs − xp wr

m

wm = −Kp xp − Kq qp

ensuring exponential stability, and for wrs = 0 we may set

ws = −Ka xa − Kp xp

wm = −Kq qp

also leading to exponential stability. Thus the situation for apredefined smooth reference trajectory is well treated. Howevertrajectory following may not be the optimal strategy whenrobots attempting rendezvous take initial positions withinmutual proximity. A reference trajectory would define a pathfrom the initial state to a desired final state including themutual rendezvous position. However the precise rendezvouspoint may not be of major significance as long as it iswithin proximity of the initial positions. Therefore we definean overall system description of two robots within mutualproximity, based on local relative coordinates.

C. Proximity Control

Let x1, x2 and q1, q2 denote the position and direction ofrobots 1 and 2. We define local coordinates e1, e2 and e3 by

e1 =< x1 + cRq1 − x2, q1 >

e2 =< x1 + cRq1 − x2, Rq1 >

e3 =< q2, Rq1 >

where c = 2w, i.e. twice the width of the robot. This yieldsthe following bilinear dynamics

e1 = w1s − cw1

m + w2s + w1

me2

e2 = −w1me1

e3 = w1m − w2

m (3)

Local coordinates e1, e2 and e3 are selected so that the origo(e1, e2, e3) = (0, 0, 0) is the desired final state of rendezvous;namely where robots are parked parallelly in a touchingconfiguration. Local relative positional coordinates e1, e2 areshown in figure(3).

Allowing both robots to move under proximity operationyields an overall holonomic system, i.e. where there are morecontrols than degrees of freedom in the system. In this casethis allows us to impose further criteria onto the controller asfor example energy considerations. As an example we may letthe robot most in need of battery move the least. Thus werestrict to w2

s = 0, that is we allow robot 2 only to perform

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Fig. 3. Local relative coordinates e1, e2.

directional change. The proposed control design is based onthe following input state linearization

w1m = −u2/e1

w2m = −u3 − w1

m

w1s = u1 + cw1

m − w1me2

so that

e1 = u1

e2 = u2

e3 = u3 (4)

we assume an initial configuration where e3 ≈ 0 and e2 −l/2 e3 > 0 as shown in figure (4) We propose a linear control

q1

Rq1q2

Rq2

e2-l/2 e3e2

Fig. 4. Initial configuration.

u1 = −K1 e1, u2 = −K2 e2, u3 = −K3 e3, where positivefeedback gains K1,K2,K3 fulfill K1 < K2 < K3. Thisensures exponential convergence and a finite w1

m as well ase2(t) − l/2 e3(t) > 0 for all t ≥ 0, where the latter prohibitscollision as indicated in figure (4). We omit further treatmentof how to obtain the assumed initial condition, but point to thelinearized system (4) to indicate the minor significance of thisproblem. Simulation results are shown in figure (5).

0 0.5 1 1.5 2 2.5 3 3.5 4−1

−0.5

0

0.5

1

1.5

2

2.5

3

Fig. 5. Convergence from initial configuration to rendezvous.

V. PROBABILISTIC MODELLING

The model developed in this section aims to combinethe effects of all the influential mechanisms associated totrophallaxis in mobile robot populations: mobility, resourcesharing, charging and resource consumption. Initially separatemodels are developed for each of the above effects, whichare then assumed additive and conditionally independent in agiven system state. System state is defined to be the positionxi(t) of every robot i, its velocity vi(t) as well as its energyresource bi(t). Since we aim for a probabilistic model, exactvalues of state variables are not tracked. Instead the developedmodel follows the evolution of the distribution of these randomvariables and in particular the distribution under stationarity.Non parametric equations are developed for distributions ofxi(t) and vi(t), whereas the conditional distribution for bi(t)is described in terms of 1 st. and 2 nd. moments. To easeexposition, velocities are assumed to take values within adiscrete set. Each separate model for resource sharing, chargingand consumption are given as integro-differential equationsgoverning the time evolution of the conditional expectationbi(t, x, v) = E(bi(t)|xi(t) = x, vi(t) = v) of the resourcebi(t) carried by robot i, given it is located at x at timet, with velocity v. Likewise a second moment model isdeveloped for the time evolution of the conditional varianceVi(t, x, v) = V AR(bi(t)|xi(t) = x, vi(t) = v).

A. Mobility

Several mobility models exist, including deterministic aswell as random movement. Among others we find the Randomwaypoint [4], Random direction and Random trip [5]. Anoverview is presented in [7]. In this work we consider a classof mobility models both suited for probabilistic modelling ofwhich a special case is Brownian motion [6]. Brownian motionis characterized by its lack of velocity persistence, i.e. unitsmove without memory of previous direction and speed.The Less Drunk Model (LDM) is as Brownian motion aMarkovian mobility model, where velocity remains constantbetween time instants {tn} of velocity change. Time instants{tn} are assumed to be a homogenous Poisson process with

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intensity parameter λ.At times {tn} of velocity change velocity is drawn indepen-dently from a distribution LQ(x(tn)). As a consequence atime dependent joint position/velocity distribution Li(A×W, t)expressing the kinematic state of robot i may be found, whichis assumed to be composed as follows

Li(A × W, t) =∫

A

Li(W |x, t) · f iL(x, t)dx

, i.e. the distribution of position possesses a density functionfL. The LDM constitutes a Markov process transformingkinematic state distribution as follows

d

dt(Li({vj}|x, t) · f i

L(x, t))

= λ(x)(LQ({vj}, x) − Li({vj}|x, t))f iL(x, t)

− < vj , Dx[Li({vj}|·, ·)f iL](x, t) > (5)

when LQ is assumed to concentrate probability on discretevelocities {vj}. Transformation (5) under mild assumptionsdefines a unique stationary solution Li({vj}|x) ·f i

L(x), which,by suitable selection of λ(x) and LQ, may be set to model anyobserved kinematic distribution of robots, e.g. to be uniformwithin D. The stationary solution constitutes the basis of theanalysis resource exchange effects in the following.

B. Resource Exchange

Energy exchange is in this work considered to be anunplanned epidemic process, i.e. transfer of energy betweenrobots take place during accidental rendezvous. All mobileunits are assumed to move randomly in patterns generatedby a Less Drunk mobility process as described above. Whentwo robots come within a suitable (not too large) distance toeach other, conditions promote energy exchange as illustratedin figure (6). The proposed exchange policy is randomizedin a way to account for both proximity and difference inresource, i.e. the nearer the robots and the higher difference,the higher probability of exchange from the higher resourceunit to the lower. More precisely two robots i and j positionedat positions xi and xj respectively are assumed to engage ina battery exchange within the time interval [t, t + dt] witha probability α dt K(xi, xj), where α is a rate parameterand K is a neighbourhood kernel modelling the dependenceof relative/absolute positions on exchange probability. Thedecision to engage in battery exchange is taken randomlyand represented by the random Boolean selector Ceij , whereP (Ceij) = α dt K(xi, xj). At time t robots i and jmutually communicate remaining battery resources bi(t) andbj(t) respectively. The final choice of battery exchange is takenrandomly and represented by the random Boolean selectorCfij , where

P (Cfij) = C|bi(t) − bj(t)|

where C is chosen, so that P (Cf ) ≤ 1. If exchange isdecided, a fixed size quantity Q is exchanged, where Q =

Fig. 6. Two robots in accidental rendezvouz, candidating for energy exchange.

|Q| sign(bj(t)− bi(t)). Altogether the exchange dynamics forrobot i can be written as

bi(t+dt) = bi(t)+∑

j

Ceij Cfij |Q| sign(bj(t)−bi(t)) (6)

where j ranges over all robots in the swarm. Taking expecta-tions conditioned position and velocity in equation (6) leadsto the desired integro differential equation, which is presentedsubsequently as a part of the complete model.

C. Charging Stations

Charging stations may be considered simply as robot unitsserving special objectives. Specific to charging stations isthe fact, that batteries should never be received by these,and additionally that they may move according to a specificmobility patterns. With respect to the former exception weexclude from the model the resource level of charging stationsand simply assume resource levels always to assume an upperbound, i.e. bi(t, x) = b ∀ t, x. This excludes the possibility ofbattery units to be handed over to charging stations. Likewise itmay be desirable to have separate control of the exchange ratefrom the charger. Thus we set the exchange rate parameter forthe charger by αC = rCα, where rC is a positive real typically> 1.

D. Energy Consumption

Various models for energy consumption in mobile roboticsare suggested in litterature [17],[18]. In this case choosinga suitable model involves a trade-off between precision andmathematical tractability. The rate of energy consumption maydepend on various parts of the system state, i.e. on aspectsof the state of the entire population as well as the stateof the individual robot. Since robots may be equipped withenergy preserving activity policies, their individual activitymay depend on their remaining energy resources. Taking suchbehaviour into account may be achieved by letting consump-tion rate depend on remaining resources. In this case we

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suggest a Poisson modulated model, i.e.

bi(t) = bi(0) · exp−n(−r/γ) for t ∈ [tn, tn+1] (7)

where {tn} is an homogenous Poisson sequence of timeinstants, where remaining battery resources are discountedthrough multiplication by exp(−r/γ) < 1. For our Poissonmodulated consumption model (7), we may deduce

d

dtbi(t, x, v) = γ (exp(−r/γ) − 1) bi(t, x, v)

d

dtb2i(t, x, v) = γ (exp(−2r/γ) − 1) b2i(t, x, v) (8)

where b2 denotes conditional second moment of the resourcebi(t).

E. Complete Model

A complete model is presented which combines the effectsof mobility, energy exchange and energy consumption. Thedeveloped model assumes the shape of integro-differentialequations governing the time evolution of the conditionalexpectation bi(t, x, v) of the battery resource bi(t) of roboti given this robot is located at position x at time t, with ve-locity v. Likewise integro-differential equations for the secondmoment b2i(t, x, v) are given.Combining various effects to the conditional expected resourceis directly accomplished by summation of terms, i.e.

f iL(x)Li({vj}|x)

d

dtbi(t, x, v)

= f iL(x)Li({vj}|x)|Q|Cα

∑j

∫D

K(x, y)bj(t, y)Lj(dy)

− f iL(x)Li({vj}|x)|Q|Cα bi(t, x, v)

∫D

K(x, y)L(dy)

− < vj , Dx[bi f iL Li](t, x, vj) >

+ λ (LQ({vj}, x) bi(t, x) − Li({vj}|x) bi(t, x, vj))f iL(x)

+ f iL(x)Li({vj}|x)γ (exp(−r/γ) − 1) bi(t, x, v) (9)

For conditional 2nd. moment further arguments are needed.The various effects on energy resource dynamics are assumedto be probabilistically independent processes. Additionally2nd. moments of changes for infinitesimally small time stepsdt are assumed to exhibit certain smoothness properties fordt = 0. Under these assumptions the various effects may becombined by summation of terms into overall dynamics for2nd. moment, i.e.

f iL(x)Li({vj}|x)

d

dtb2i(t, x, v) =

2|Q|Cf iL(x)Li({vj}|x) λ (bi(t, x)

∫Ω

K(x, y)b(t, y)L(dy)

− b2i(t)∫

Ω

K(x, y)L(dy))

+ |Q|Cf iL(x)Li({vj}|x) λ

∫Ω

K(x, y)|b(t, y) − bi(t, x)|L(dy)

− < vj , Dx[b2i f iL Li](t, x, vj) >

+ λ (LQ(vj , x) b2i(t, x) − Li({vj}|x) b2i(t, x, vj))f iL(x)

+ f iL(x)Li({vj}|x)γ (exp(−2r/γ) − 1) b2i(t, x, v) (10)

F. Numerical Example

We present a one dimensional example, where robot areassumed to move within a domain of operation D = [−1, 1].Two discrete velocities {−1, 1} are assumed with equal proba-bility 1/2 wihtin D. At positions x < −1 only velocity v = 1may be assumed and for x > 1 only v = −1. Letting λ(x)assume a high value outside D yields approximate uniformposition distribution concentrated within D.Crossing the entire domain D without speed changes lasts 2time units. For the mobility parameter λ we assume robotsto change velocity 10 times for each such 2 time units, i.e.λ = 5. In order for an energy propagation mechanism to beworthwhile, a significant power loss should be associated withtravelling from the peripheral of the domain of operation to thecharger. Thus we assume, that a direct travel half way acrossD discounts the energy resources by 2/3, i.e. exp(−r) = 0.3or r ≈ 1. Regarding energy exchange, we normalize thecharger resource by b = 1 defining an upper bound for bi.In accordance we set C = 1 and Q = 1/10, that is, the energyquantum exchanged is far lower than the upper bound forremaining resource. The neighbourhood kernel K is assumedto allow energy exchange within a fixed distance d = 0.1,i.e. K(x, y) = I|x−y|<0.1. The charging process is assumed tobe faster than the energy consumption process. Thus we setγ < αC = 40. The mutual robot exchange rate α is varied toillustrate its effect on energy distribution. A charger placed ata fixed location xC = 0 serves N = 50 robots. Figures (7)and (8) show stationary energy distributions for values of α,0.2 and 10.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

N:50,xC

:0,lambda:20,alpha:0.2,gamma:20,r:1

Position

Ene

rgy

reso

urce

expectation bb+stdevb−stdev

Fig. 7. Energy distributions for low level of α = 0.2.

VI. CONCLUSION

The concept of robot trophallaxis is presented inspired bycollaboration and altruism in natural groups. A number ele-mentary problems such as energy containment, motion control,navigation, morphology and exchange policy are identified.Morphology is examplified by the CISSBot, designed to facili-tate exchange of batteries. A collision free proximity controller

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−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

N:50,xC

:0,lambda:20,alpha:10,gamma:20,r:1

Position

Ene

rgy

reso

urce

expectation bb+stdevb−stdev

Fig. 8. Energy distributions for high level of α = 10.

is presented to allow rendezvous of two robots engaged inenergy exchange. The controller is based on the morphologyof the CISSBot. An overall probabilistic model is constructedaccounting for individual behaviour effect on the overall groupenergy distribution. Effects from mobility, energy exchange,charging and energy consumption are included in the model.The model parameterizes the level of altruism among robotsby a rate parameter λ, which is illustrated by one dimensionalexamples for low and high values of λ. Examples illustratehow increased altruistic behaviour levels energy resourcesthroughout the entire working domain and decreases variabilityamong robots, which in turn improves survivability.Future research efforts include further developments of theCISSBot as well as long lasting, large scale experimentsincluding a high number of robots. Numerical computations fortwo dimensional settings have been prepared and systematicstudies of the impact of individual behaviour to group stateconstitute a focus area in the near future.

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