Discrete Bee Dance Algorithms for Pattern Formation on the Grid Noam Gordon Israel A. Wagner Alfred M. Bruckstein Technion IIT, Israel

Introduction Distributed Multi-A(ge)nt Robotics (MAR): Applying a distributed design approach to robotic systems. Multiple mobile autonomous robots work in parallel. MAR systems enjoy the inherent advantages of distributed systems: Increased performance (parallelism, load balancing, adaptivity, etc.); Increased performance (parallelism, load balancing, adaptivity, 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 advanced logic, sensors, motors and comm. Devices). Simple compact design and low cost (We may give up things like advanced logic, sensors, motors and comm. Devices).

Formation and Agreement MAR may often need to create spatial formations, e.g. to encircle or carry an object, form a sensor array, gather in a small region, etc. MAR must coordinate their actions and agree on basic things such as a reference coordinate system. When communication devices are absent, Agreement, Coordination and Formation problems become entangled.

Related works Suzuki, Yamashita et al (’93, ’96). Prencipe, Flocchini et al (’00, ’01, ’02). Defago et al (’02). Schlude (‘02). Notable others include Arkin, Beni and Wang, Mataric.

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

The problem and our solution The problem: Make the robots arrange themselves in a desired spatial pattern (a list of coordinates q 1,…,q n ). Our solution: The agents gather in a single cell and make it their origin. The agents gather in a single cell and make it their origin. The agents agree on the x and y axes and give each agent a unique id. The agents agree on the x and y axes and give each agent a unique id. Each agent simply moves to its designated destination, and the formation is created. Each agent simply moves to its designated destination, and the formation is created.

Gathering in a single cell

Each agent moves towards the Center of Mass (COM) of all agents' positions. Proof idea: In 1D, the algorithm works because agents in the extremes always move inwards. In 1D, the algorithm works because agents in the extremes always move inwards. The algorithm works in 1D even with crash failures (i.e., if some agents “die”); The “living” agents still gather. The algorithm works in 1D even with crash failures (i.e., if some agents “die”); The “living” agents still gather. In 2D, we observe the projection of the system on each axis, and the problem reduces to 1D with crash failures. In 2D, we observe the projection of the system on each axis, and the problem reduces to 1D with crash failures. The agents coordinate their gathering and agreement on origin, by raising “flags” which are visible only from within the same cell (This is a model for “touch”).

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

Breaking symmetries with Strong Asynchronicity Suzuki et al and others pointed out that agents may be unable to break symmetries if they are synchronous. As a result, only symmetric patterns are feasible, and agreement on axis direction cannot be achieved. We believe that total synchrony is unlikely in reality. Thus, symmetries can be easily broken using the random asynchronicity of the agents. We guarantee the eventual symmetry breaking using the Strong Asynchronicity assumption: For each group of agents G and at any time step t, there is a chance that G is the group of waking (active) agents at time t. For each group of agents G and at any time step t, there is a chance that G is the group of waking (active) agents at time t. This assumption ensures that all symmetries are eventually broken, including the possible ties in the gathering and voting procedures above.

Agreement on a total ordering Here we demonstrate how the agents exchange numerical information through their “dances”. Each agent moves, along a specific path, to a location which is a function of its initial location. We assume that, initially, the agents occupied distinct locations, so their initial positions can be sorted and used for ordering the agents. Once all agents assume their proper locations, each agent is able to calculate its own unique id.

Agreement on a total ordering

Formation of the pattern Now that the agents have a common coordinate system and unique ids, they can easily form any pattern: Given a desired pattern (q 1, …, q n ), agent i moves directly to location q i. Given a desired pattern (q 1, …, q n ), agent i moves directly to location q i.

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