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

2. Agenda Introduction Spring network Rotating leadership election Future research

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

4. 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.

5. 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

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

7. 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.

8. 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.

9. 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.

10. Multiple Leaders

11. 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.

12. 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.

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

14. 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.

15. Non-leader motion

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

17. 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.

18. 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

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

20. 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

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

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

23. 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.

24. 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.

25. 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…

26. 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

27. 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.

28. 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.

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

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

31. THANK YOU