Discrete Bee Dance Discrete Bee Dance Algorithms for Pattern Algorithms for Pattern Formation on the GridFormation on the Grid

Noam GordonNoam GordonIsrael A. WagnerIsrael A. Wagner

Alfred M. BrucksteinAlfred M. Bruckstein

Technion IIT, IsraelTechnion IIT, Israel

IntroductionIntroduction

Distributed Multi-A(ge)nt Robotics (MAR): Applying a Distributed Multi-A(ge)nt Robotics (MAR): Applying a distributed design approach to robotic systems.distributed design approach to robotic systems.Multiple mobile autonomous robots work in parallel.Multiple mobile autonomous robots work in parallel.MAR systems enjoy the inherent advantages of MAR systems enjoy the inherent advantages of distributed systems:distributed systems:

Increased performance (parallelism, load balancing, adaptivity, Increased performance (parallelism, load balancing, adaptivity, etc.);etc.);

Robustness. No critical element or “weak link”;Robustness. No critical element or “weak link”; Simple compact design and low cost (We may give up things like Simple compact design and low cost (We may give up things like

advanced logic, sensors, motors and comm. Devices).advanced logic, sensors, motors and comm. Devices).

Formation and AgreementFormation and Agreement

MAR may often need to create spatial MAR may often need to create spatial formations, e.g. to encircle or carry an object, formations, e.g. to encircle or carry an object, form a sensor array, gather in a small region, form a sensor array, gather in a small region, etc.etc.

MAR must coordinate their actions and agree on MAR must coordinate their actions and agree on basic things such as a reference coordinate basic things such as a reference coordinate system.system.

When communication devices are absent, When communication devices are absent, Agreement, Coordination and Formation Agreement, Coordination and Formation problems become entangled.problems become entangled.

Related worksRelated works

Suzuki, Yamashita et al (’93, ’96).Suzuki, Yamashita et al (’93, ’96).

Prencipe, Flocchini et al (’00, ’01, ’02).Prencipe, Flocchini et al (’00, ’01, ’02).

Defago et al (’02).Defago et al (’02).

Schlude (‘02).Schlude (‘02).

Notable others include Arkin, Beni and Notable others include Arkin, Beni and Wang, Mataric.Wang, Mataric.

Our modelOur model

Our robots (or agents) are Our robots (or agents) are nn points on the rectangular points on the rectangular grid.grid.We focus on minimalist We focus on minimalist anonymousanonymous and and homogeneoushomogeneous agents which have agents which have no reference coordinate systemno reference coordinate system and and no tele-communicationno tele-communication devices. They shall reach devices. They shall reach agreement and exchange information by observing each agreement and exchange information by observing each other's movements, hence the term ``Bee Dance other's movements, hence the term ``Bee Dance Algorithms''.Algorithms''.Time is discrete.Time is discrete.Asynchronicity: In each time step, each robot may or Asynchronicity: In each time step, each robot may or may not wake up and move to one of four adjacent cells.may not wake up and move to one of four adjacent cells.A robot may sleep for an indefinite time, but will A robot may sleep for an indefinite time, but will eventually wake up.eventually wake up.

The problem and our solutionThe problem and our solution

The problemThe problem: Make the robots arrange : Make the robots arrange themselves in a desired spatial pattern (a themselves in a desired spatial pattern (a list of coordinates list of coordinates qq11,…,q,…,qnn).).

Our solutionOur solution:: The agents gather in a single cell and make it The agents gather in a single cell and make it

their origin.their origin. The agents agree on the The agents agree on the xx and and yy axes and axes and

give each agent a unique give each agent a unique idid.. Each agent simply moves to its designated Each agent simply moves to its designated

destination, and the formation is created.destination, and the formation is created.

Gathering in a single cellGathering in a single cell

Gathering in a single cellGathering in a single cell

Each agent moves towards the Each agent moves towards the Center of MassCenter of Mass (COM) of (COM) of all agents' positions.all agents' positions.Proof idea:Proof idea:

In 1D, the algorithm works because agents in the extremes In 1D, the algorithm works because agents in the extremes always move inwards.always move inwards.

The algorithm works in 1D even with crash failures (i.e., if some The algorithm works in 1D even with crash failures (i.e., if some agents “die”); The “living” agents still gather.agents “die”); The “living” agents still gather.

In 2D, we observe the In 2D, we observe the projectionprojection of the system on each axis, and of the system on each axis, and the problem reduces to 1D with crash failures.the problem reduces to 1D with crash failures.

The agents coordinate their gathering and agreement on The agents coordinate their gathering and agreement on origin, by raising “flags” which are visible only from within origin, by raising “flags” which are visible only from within the same cell (This is a model for “touch”).the same cell (This is a model for “touch”).

Agreement on the axesAgreement on the axes

Each agent “votes” on a Each agent “votes” on a preferred direction by moving preferred direction by moving one step from the origin.one step from the origin.The most “popular” direction The most “popular” direction becomes the reference axis becomes the reference axis direction.direction.In case of a tie, agents change In case of a tie, agents change their votes, until the tie is their votes, until the tie is broken.broken.The agents acknowledge the The agents acknowledge the choice by moving another step choice by moving another step outwards and then returning to outwards and then returning to the origin.the origin.

Breaking symmetries with Strong Breaking symmetries with Strong AsynchronicityAsynchronicity

Suzuki et al and others pointed out that agents may be Suzuki et al and others pointed out that agents may be unable to break symmetries if they are synchronous.unable to break symmetries if they are synchronous.As a result, only symmetric patterns are feasible, and As a result, only symmetric patterns are feasible, and agreement on axis direction cannot be achieved.agreement on axis direction cannot be achieved.We believe that total synchrony is unlikely in reality.We believe that total synchrony is unlikely in reality.Thus, symmetries can be easily broken using the Thus, symmetries can be easily broken using the random asynchronicity of the agents.random asynchronicity of the agents.We guarantee the eventual symmetry breaking using the We guarantee the eventual symmetry breaking using the Strong AsynchronicityStrong Asynchronicity assumption: assumption:

For each group of agents For each group of agents GG and at any time step and at any time step tt, there is a , there is a chance that chance that GG is the group of waking (active) agents at time is the group of waking (active) agents at time tt..

This assumption ensures that all symmetries are This assumption ensures that all symmetries are eventually broken, including the possible ties in the eventually broken, including the possible ties in the gathering and voting procedures above.gathering and voting procedures above.

Agreement on a total orderingAgreement on a total ordering

Here we demonstrate how the agents exchange Here we demonstrate how the agents exchange numerical information through their “dances”.numerical information through their “dances”.

Each agent moves, along a specific path, to a Each agent moves, along a specific path, to a location which is a function of its location which is a function of its initial locationinitial location..

We assume that, initially, the agents occupied We assume that, initially, the agents occupied distinct locations, so their initial positions can be distinct locations, so their initial positions can be sorted and used for ordering the agents.sorted and used for ordering the agents.

Once all agents assume their proper locations, Once all agents assume their proper locations, each agent is able to calculate its own unique each agent is able to calculate its own unique idid..

Agreement on a total orderingAgreement on a total ordering

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Formation of the patternFormation of the pattern

Now that the agents Now that the agents have a common have a common coordinate system coordinate system and unique ids, they and unique ids, they can easily form any can easily form any pattern:pattern: Given a desired Given a desired

pattern pattern (q(q11, …, q, …, qnn)), ,

agent agent ii moves directly moves directly to location to location qqii..

ConclusionConclusion

We have shown how agents can coordinate, agree and We have shown how agents can coordinate, agree and exchange information through “bee dance” algorithms exchange information through “bee dance” algorithms based only on movement and observation, without using based only on movement and observation, without using explicit data communication.explicit data communication.We have demonstrated the intimate relationship between We have demonstrated the intimate relationship between agreement and formation problems with such agreement and formation problems with such communication-less robots.communication-less robots.We have suggested a formal notion of the asynchronous We have suggested a formal notion of the asynchronous nature of autonomous mobile agents, and demonstrated nature of autonomous mobile agents, and demonstrated its power to break symmetries and enable the agents to its power to break symmetries and enable the agents to reach agreement.reach agreement.