Gathering Multiple Robotic A(ge)nts with limited visibility Noam Gordon Israel A. Wagner Alfred M. Bruckstein Technion - IIT

The Gathering Problem  How to make multiple autonomous robots gather in a small region/point?  a.k.a. Point Formation or Convergence.  Fundamental to formation and self- organization problems.  Useful for collecting robots after a mission or after being initially dispersed.  Useful for nano-robot aggregation.

Current Works  Suzuki et al. ’96–’99  Prencipe et al. ’01–’03  Bruckstein et al. ’91–’03  Francis et al. ’03–’04

The World Model  The agents are points in the plane. points in the plane. “semi-synchronous” – moving in synchronous steps, but randomly scheduled to act only during some steps. “semi-synchronous” – moving in synchronous steps, but randomly scheduled to act only during some steps. anonymous, homogeneous, memoryless. anonymous, homogeneous, memoryless. able to move up to a distance σ in one step. able to move up to a distance σ in one step.  An agent can see only up to a distance V.  An agent cannot measure the distance, but rather only the direction toward a nearby agent.

Maintaining Visibility abV

a V

a

 The intersection of circles is empty.  The agent is “surrounded” and cannot move. a

The Proposed Algorithm a ψ

 Move as far as possible in the direction of the bisector of ψ, inside the allowable region.  Based on Suzuki and Sugihara’s algorithm, with the addition of step size control.  Step length =

The proposed algorithm  The idea is that agents tend to move closer to each other while maintaining visibility.  The agents at the outskirts move inside, making the “cloud” of agents contract.  The visibility graph has nodes for agents and edges for mutual visibility of agents.  If the visibility graph is connected, all agents will gather in a small cluster.

Emerging Behavior – The Contraction Phase  All agents contract to a small region.  During the process, the shape of the occupied region becomes an approximate polygon.  The polygon “corners” are dense clusters of many agents.  We believe there is a positive feedback loop between the boundary “curvature” and agent density along the boundary.

Curvature ↔ Density  The density of agents along the boundary affects its curvature and vice versa. High density means that more agents need to leap one over the other, so the local contraction is slower on average and curvature rises. High density means that more agents need to leap one over the other, so the local contraction is slower on average and curvature rises. High curvature means that the agents often move closer to each other, so the density rises. High curvature means that the agents often move closer to each other, so the density rises.

Emerging Behavior – The Wandering phase  Once inside a region of diameter ~ O(σ), the cluster ceases to contract.  The outermost agents leap over the cluster rather than enter it, because of their relatively large steps.  The cluster is now a composite random walker!  Conjecture: If several disconnected clusters exist, they will all eventually merge.

The Continuous-time Analog  If we let σ=v∆t, then in the limit ∆t→0, the algorithm becomes: If not surrounded, then move along the bisector of ψ at a constant speed v.  Surprisingly, the agents may actually move at varying speeds, unequal to v! How come?

Collinearity → Zenoness → Varying Speeds

 The middle agent b of 3 collinear agents a,b,c will constantly “try” to remain collinear: As agents a,c move, agent b chases the segment ac and eventually crosses it. As agents a,c move, agent b chases the segment ac and eventually crosses it. Agent b turns back, crosses ac again, and so forth. Agent b turns back, crosses ac again, and so forth. Thus, a chattering movement occurs, arbitrarily close to the segment ac. Thus, a chattering movement occurs, arbitrarily close to the segment ac.

Collinearity → Zenoness → Varying Speeds  The agent exhibits Zeno behavior – an infinite amount of switches in finite time.  The chattering movement translates to a seemingly (and arbitrarily) smooth motion on ac, at a speed dependent on the other agents’ movements, and generally unequal to v.

Contraction and Shape Evolution  The middle agent(s) in a collinear group of agents do not affect the corner agents.  Agent don’t leap one over the other.  Therefore, the local density does not affect local contraction speed.  There is no Curvature ↔ Density positive feedback here!  An approximate polygon will probably not be formed in the continuous-time case!  The agents will gather in a single point within finite time. The will be no wandering!

Conclusion  We have shown how to gather simple robots with very limited visibility.  Interesting global phenomena occur – large-scale polygon formation and a randomly walking cluster.  In the continuous-time analog, these phenomena do not occur.  We have also presented a discrete-space analog (on the rectangular grid).

Further Work  Robust analysis of the system evolution: Boundary shape evolution; Boundary shape evolution; Movement and merging of wandering clusters. Movement and merging of wandering clusters.  Noise, errors and delay.  Non-point robots and collisions.  Other sensing models.  Alternative movement algorithms (e.g., random).  Formation of other (convex) shapes.