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Direction Election in Flocking Swarms Ohad Ben-Shahar, Shlomi Dolev Andrey Dolgin, Michael Segal Ben-Gurion University of the Negev

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Agenda Introduction Spring network Rotating leadership election Future research

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Introduction Gathering and Leading Direction Election Entities obtain only position of neighbors Multiple entities may want to lead What problem we are dealing with?

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Agent model Agent dynamics Control laws Linear (Vicsek) Nonlinear (Reynolds) Why is it exponentially decreasing with time??? ANALYSIS OF COORDINATION IN MULTI-AGENT SYSTEMS THROUGH PARTIAL DIFFERENCE EQUATIONS. PART II: NONLINEAR CONTROL Giancarlo Ferrari-Trecate, Annalisa Bua,Mehdi Gati, IFAC 2005.

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Reynolds Rules http://www.red3d.com/cwr/boids/ Reynolds suggests three intuitive rules : Separation Alignment Cohesion steer towards the average heading of local flockmates steer to avoid crowding local flockmates steer to move toward the average position of local flockmates

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Reynolds Rules Unfortunately do not cope with Symmetry Two move in opposite directions or towards each other preventing simultaneous flocking

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Related Work [TJP03] Potential function: Collision avoidance Maintaining links Single leader only No errors Tanner, H. G., Jadbabaie, A., and Pappas, G. J., Stable flocking of mobile agents, Part II: dynamic topology", Proc. IEEE Conference on Decision and Control, Maui, Hawaii, pp. 2016-2021, 2003.

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Related Work [TJP03] IF the graph is always connected then All pair-wise velocity differences converge asymptotically to zero, Collisions between the agents are avoided. Tanner, H. G., Jadbabaie, A., and Pappas, G. J., Stable flocking of mobile agents, Part II: dynamic topology", Proc. IEEE Conference on Decision and Control, Maui, Hawaii, pp. 2016-2021, 2003.

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Related Work [JW09] Multiple Leaders All entities exponentially converge to the weighted average position and velocity of leaders, WHEN connectivity is preserved. Jiang-Ping, H., Hai-Wen, Y., Collective coordination of multi-agent systems guided by multiple leaders", IEEE Transactions On Robotics, Vol. 18, No. 9, 2009.

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Multiple Leaders

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Our Spring Network Definition Spring is a virtual structure connecting any two neighboring entities. The force that the spring applies on its ends is F = (r ij - (R - r)/2)/2. The spring attains its equilibrium state in the middle between R and r. Moreover a spring never exceeds R nor reduced to less than r.

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Spring Network Theorem: Connectivity Preserving and Collision Avoidance Given the spring graph initial connectivity And the fact that the algorithm does not violate the spring definition.

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Leader motion e includes a random variable term added for symmetry breaking

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Non Leader Motion Try to move to relax the spring Avoiding moves that may violate R or r Take in account the movement of the neighbors and the errors in the measurements.

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Non-leader motion

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Rotating Leader Election Each candidate tries to define the direction for T time.

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Rotating Leader Election A part of T is dedicated for spring network convergence --- to provide each leader a possibility to move, Since the equilibrium state of each spring is in the middle between r and R, this is the optimal position, to move in any desired direction.

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Leader Election for Labeled Entities ORDER = [1..n]. Wait until ORDER == (T global mod nT )+ 1 then lead for time slot T, preserving spring definition. Update neighbor list with newly created springs. Start over again

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Leader Election for Labeled Entities convergenceleading entity i entity i+1 waiting convergenceleadingwaiting

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Leader Election for Unlabeled Entities Uniformly choose ORDER on the range [1,P]. Wait until ORDER == (T global mod P )+ 1 then lead for time slot T, preserving spring definition. Update neighbor list with newly created springs Start over again

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Leader Election for Unlabeled Entities convergenceleading entity i entity j waiting convergence leadingwaitingconvergence leading

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Leader Election The probability for all entities to have a chance to lead alone for time slot T

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Leader election theorem Theorem: Direction election algorithms will make the swarm follow a single leader at least k times in a leading period with predetermined probability.

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Leader election for unsynchronized clocks An additional part in the time slot T should be allocated for synchronization. Allowing entities to lead only after this additional part, no two entities with different ORDER values can compete for leadership. All other properties are preserved.

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Leader election for unsynchronized clocks ORDER=k convergence synchronization leading entity i waiting ORDER=k+1 convergence synchronization leading entity j waiting start s with j T length period If starts more than T/2 following the previous and more than T/2 before the next, then no collision, same probability as having an index of T for yourself in the slotted T (rather than 2T) case

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Leader election for unsynchronized clocks Uniformly choose ORDER in the range [1,P]. Wait until ORDER == (T global mod P )+ 1 then lead for time slot T, preserving spring definition. Update neighbor list with newly created springs Start over again

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Leader election asynchronized clocks The probability for all entities to have a chance to lead alone is the same, since entities with different ORDER lead in different time slots.

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Leader election Priority Scheduling N P different leading scenarios are possible. N P =1 for highest priority. ORDER is multiplied by N P. Leader Election Algorithm.

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Calculating T F L The relative error is bounded by 2X/L goes down as L increases. X is bounded by nR. X 2X

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Conclusion Bounded errors are considered. Collision avoidance and connectivity is preserved all the time. Leadership direction is efficiently elected with predetermined probability.

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THANK YOU