1. A Randomized Gathering Algorithm for Multiple Robots with Limited Sensing Capabilities Noam Gordon Israel A. Wagner Alfred M. Bruckstein Technion – Israel Institute of Technology

2. 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.

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

4. The World Model The agents are  points in the plane.  “semi-synchronous” – moving in synchronous steps, but randomly scheduled to act only during some steps.  anonymous, homogeneous, memoryless.  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.

5. Maintaining Visibility abV

6. a V

7. a

8. The intersection of circles is empty. The agent is “surrounded” and cannot move. If there are no visible agents, then the allowable region is a full disc. a

9. A deterministic algorithm In previous work (‘04), we proposed a deterministic algorithm. Related to an algorithm by Sugihara et al (‘96). a

10. The Proposed Randomized Algorithm a

11. The agents simply move randomly while maintaining visibility. Similar global behavior remains.  Phase 1: Non-uniform contraction of the swarm, assuming a rough large-scale polygonal shape, ultimately becoming a small dense cluster.  Phase 2: The cluster stops contracting and begins wandering in the plane.

12. Proof of convergence Lemma 1: If only a 1 is active and it moves a step s closer to COM, then the variance ∑d i 2 will decrease by at least |s| 2 /n. a1a1 d1d1 s θ

13. Proof of convergence d = (d 1,…,d n ) T = displacement from COM d T d is the variance. s = (s,0,…,0) T = agents’ steps (only a 1 moves) d’ = displacement from COM after move. 1 = (1,…,1) T d’ = d – s + 1·s/n

14. Proof of convergence d’ = d – s + 1·s/n d’ T d’ = (d – s + 1·s/n) T (d – s + 1·s/n) = d T d + s(s-2d) – s T s/n = d T d + |s|(2|d 1 |cosθ-|s|)-|s| 2 /n a 1 moves closer to COM iff the second term is negative. In this case, d’ T d’ - d’ T d’ ≤ -|s| 2 /n

15. Proof of convergence Lemma 2: There exists a vertex a i in CH with bounded angle φ i ≤ φ* 0 from the center of mass. Denote w.l.o.g. a 1 = a i. aiai d i ≥ d* > 0 φ i ≤ φ* < π

16. Proof of convergence Lemma 3: There exist p*,s* > 0, such that a 1 will choose to move a step of length > s*, closer to COM, with probability > p*. a1a1 COM ∂CH

17. Proof of convergence Theorem: The agents will gather and remain forever within a cluster of diameter < V, given a connected initial visibility graph. Proof: From above lemmas, there always exists an agent a 1, such that if it is the only active agent, then the variance will decrease by at least s* 2 /n, with probability p*. From our strong asynchronicity assumption, the probability that indeed a 1 will be the only active agent, is at least some ε > 0.

18. Proof of convergence Thus, there is always a probability > εp* that the variance will decrease by at least s* 2 /n. The variance is bounded from above. Therefore, it will decrease arbitrarily with probability 1 and within finite expected time. The diameter will decrease along with the variance. Once it reaches V (=full visibility), it will remain bounded by V, since the algorithm maintains visibility.

19. The Wandering Cluster Once the swarm has contracted into a small dense cluster, it begins to wander in the plane. The cluster is so small, that the outer agents (at the corners of CH) most often leap over the cluster instead of moving inside it. The random scheduling and choices make the cluster wander randomly.

20. The Wandering Cluster Is the cluster’s random walk recurrent? We need to check whether the resulting random walk is biased or not. n=1: A solitary agent performs a uniform random walk by definition. n=2: Due to symmetry, the relative orientation of two agents is itself recurrent. Hence, initial orientation is forgotten and the random walk becomes unbiased. n≥3: We conjecture that the cluster’s random walk is indeed recurrent.

21. The Wandering Cluster If our conjecture is true, then the algorithm will work for all initial conditions! Each connected component of the initial visibility graph will contract to an independent cluster. All clusters will eventually merge, due to their recurrence.

22. Conclusions A simple yet powerful randomized gathering algorithm for myopic robots which cannot measure distances. A vivid example of how very simple local behaviors turn into complex global behaviors. Interestingly, the agents just move randomly while keeping visibility. Maintain visibility → Gather So is this the only thing that these humble agents can do?