Gathering Multiple Gathering Multiple Robotic A(ge)nts with Robotic A(ge)nts with

limited visibilitylimited visibility

Noam GordonNoam GordonIsrael A. WagnerIsrael A. Wagner

Alfred M. BrucksteinAlfred M. BrucksteinTechnion - IITTechnion - IIT

The Gathering ProblemThe Gathering Problem

How to make multiple autonomous robots How to make multiple autonomous robots gather in a small region/point?gather in a small region/point?

a.k.a. a.k.a. Point FormationPoint Formation or or ConvergenceConvergence.. Fundamental to formation and self-Fundamental to formation and self-

organization problems.organization problems. Useful for collecting robots after a mission Useful for collecting robots after a mission

or after being initially dispersed.or after being initially dispersed. Useful for nano-robot aggregation.Useful for nano-robot aggregation.

Current WorksCurrent Works

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

The World ModelThe World Model The agents are The agents are

points in the plane.points in the plane. ““semi-synchronous” – moving in synchronous semi-synchronous” – moving in synchronous

steps, but randomly scheduled to act only during steps, but randomly scheduled to act only during some steps.some steps.

anonymous, homogeneous, memoryless.anonymous, homogeneous, memoryless. able to move up to a distance able to move up to a distance σσ in one step. in one step.

An agent can see only up to a distance An agent can see only up to a distance VV.. An agent An agent cannot measure the distancecannot measure the distance, but , but

rather rather only the directiononly the direction toward a nearby toward a nearby agent.agent.

Maintaining VisibilityMaintaining Visibility

a b V

Maintaining VisibilityMaintaining Visibility

a V

Maintaining VisibilityMaintaining Visibility

a

Maintaining VisibilityMaintaining Visibility

The intersection of The intersection of circles is empty.circles is empty.

The agent is The agent is “surrounded” and “surrounded” and cannot move.cannot move. a

The Proposed AlgorithmThe Proposed Algorithm

a

ψ

The Proposed AlgorithmThe Proposed Algorithm

Move as far as possible in the direction of Move as far as possible in the direction of the bisector of the bisector of ψψ, inside the allowable , inside the allowable region.region.

Based on Suzuki and Sugihara’s Based on Suzuki and Sugihara’s algorithm, with the addition of step size algorithm, with the addition of step size control.control.

Step length = Step length = ),2

cos,2

min( V

V

The proposed algorithmThe proposed algorithm

The idea is that agents tend to move The idea is that agents tend to move closer to each other while maintaining closer to each other while maintaining visibility.visibility.

The agents at the outskirts move inside, The agents at the outskirts move inside, making the “cloud” of agents contract.making the “cloud” of agents contract.

The The visibility graphvisibility graph has nodes for agents has nodes for agents and edges for mutual visibility of agents.and edges for mutual visibility of agents.

If the visibility graph is connected, all If the visibility graph is connected, all agents will gather in a small cluster.agents will gather in a small cluster.

Emerging Behavior – Emerging Behavior – The The Contraction PhaseContraction Phase

All agents contract to a small region.All agents contract to a small region. During the process, the shape of the During the process, the shape of the

occupied region becomes an approximate occupied region becomes an approximate polygon.polygon.

The polygon “corners” are dense clusters The polygon “corners” are dense clusters of many agents.of many agents.

We believe there is a positive feedback We believe there is a positive feedback loop between the boundary “curvature” loop between the boundary “curvature” and agent density along the boundary.and agent density along the boundary.

Curvature ↔ DensityCurvature ↔ Density

The density of agents along the boundary The density of agents along the boundary affects its curvature and vice versa.affects its curvature and vice versa. High density means that more agents need to High density means that more agents need to

leap one over the other, so the local leap one over the other, so the local contraction is slower on average and contraction is slower on average and curvature rises.curvature rises.

High curvature means that the agents often High curvature means that the agents often move closer to each other, so the density move closer to each other, so the density rises. rises.

Emerging Behavior – Emerging Behavior – The The Wandering phaseWandering phase

Once inside a region of diameter ~ O(Once inside a region of diameter ~ O(σσ), the ), the cluster ceases to contract.cluster ceases to contract.

The outermost agents leap over the cluster The outermost agents leap over the cluster rather than enter it, because of their relatively rather than enter it, because of their relatively large steps.large steps.

The cluster is now a composite random walker!The cluster is now a composite random walker! Conjecture: If several disconnected clusters Conjecture: If several disconnected clusters

exist, they will all eventually merge.exist, they will all eventually merge.

The Continuous-time AnalogThe Continuous-time Analog

If we let If we let σσ==v∆tv∆t, then in the limit , then in the limit ∆t∆t→0, the →0, the algorithm becomes:algorithm becomes:If not surrounded, then move along the If not surrounded, then move along the bisector of bisector of ψψ at a constant speed vat a constant speed v..

Surprisingly, the agents may actually Surprisingly, the agents may actually move move at varying speedsat varying speeds, unequal to , unequal to vv! ! How come?How come?

Collinearity → Zenoness → Varying Collinearity → Zenoness → Varying SpeedsSpeeds

Collinearity → Zenoness → Varying Collinearity → Zenoness → Varying SpeedsSpeeds

The middle agent The middle agent bb of 3 collinear agents of 3 collinear agents a,b,ca,b,c will constantly “try” to remain will constantly “try” to remain collinear:collinear: As agents As agents a,ca,c move, agent move, agent bb chases the chases the

segment segment ac ac and eventually crosses it.and eventually crosses it. Agent b turns back, crosses Agent b turns back, crosses acac again, and so again, and so

forth.forth. Thus, a chattering movement occurs, Thus, a chattering movement occurs,

arbitrarily close to the segment arbitrarily close to the segment acac..

Collinearity → Zenoness → Varying Collinearity → Zenoness → Varying SpeedsSpeeds

The agent exhibits The agent exhibits ZenoZeno behavior – an behavior – an infinite amount of switches in finite time.infinite amount of switches in finite time.

The chattering movement translates to a The chattering movement translates to a seemingly (and arbitrarily) smooth motion seemingly (and arbitrarily) smooth motion on on acac, at a speed dependent on the other , at a speed dependent on the other agents’ movements, and generally agents’ movements, and generally unequal to unequal to vv..

Contraction and Shape EvolutionContraction and Shape Evolution

The middle agent(s) in a collinear group of The middle agent(s) in a collinear group of agents do not affect the corner agents.agents do not affect the corner agents.

Agent don’t leap one over the other.Agent don’t leap one over the other. Therefore, the local density does not affect local Therefore, the local density does not affect local

contraction speed.contraction speed. There is no Curvature ↔ Density positive There is no Curvature ↔ Density positive

feedback here!feedback here! An approximate polygon will probably An approximate polygon will probably notnot be be

formed in the continuous-time case!formed in the continuous-time case! The agents will gather in a single point within The agents will gather in a single point within

finite time. The will be finite time. The will be no wanderingno wandering!!

ConclusionConclusion

We have shown how to gather simple We have shown how to gather simple robots with very limited visibility.robots with very limited visibility.

Interesting global phenomena occur – Interesting global phenomena occur – large-scale polygon formation and a large-scale polygon formation and a randomly walking cluster. randomly walking cluster.

In the continuous-time analog, these In the continuous-time analog, these phenomena do not occur.phenomena do not occur.

We have also presented a discrete-space We have also presented a discrete-space analog (on the rectangular grid).analog (on the rectangular grid).

Further WorkFurther Work

Robust analysis of the system evolution: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.Noise, errors and delay. Non-point robots and collisions.Non-point robots and collisions. Other sensing models.Other sensing models. Alternative movement algorithms (e.g., random).Alternative movement algorithms (e.g., random). Formation of other (convex) shapes.Formation of other (convex) shapes.