Squaring the Circle with Weak Mobile Robots

The Circle Formation Problem (CFP) consists in the design of a protocol insuring that starting from an initial arbitrary configuration (where no two robots are at the same position), n robots eventually form a regular n -gon in finite time. None of the deterministic solutions proposed so far works for a system made of 4 or 3 robots. This comes from the fact that it is very difficult to maintain a geometric invariant with such a few number of robots, e.g., the smallest enclosing circle, concentric cycles, properties of the convex hull, or a leader. As a matter of fact, due to the high rate of symmetric configurations, the problem was suspected to be unsolvable with 4 robots. In this paper, we disprove this conjecture. We present two non-trivial deterministic protocols that solves CFP with 4 and 3 robots, respectively. The proposed solutions do not require that each robot reaches its destination in one atomic step. Our result closes CFP for any number n ( > 0) of robots in the semi-synchronous model.

[1]  Xavier Défago,et al.  Circle formation for oblivious anonymous mobile robots with no common sense of orientation , 2002, POMC '02.

[2]  Xavier A. Debest,et al.  Remark About Self-Stabilizing Systems , 1995, Commun. ACM.

[3]  Masafumi Yamashita,et al.  Erratum: Distributed Anonymous Mobile Robots: Formation of Geometric Patterns , 2006, SIAM J. Comput..

[4]  Nicola Santoro Distributed Algorithms for Autonomous Mobile Robots , 2006, IFIP TCS.

[5]  Ronald L. Graham,et al.  An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set , 1972, Inf. Process. Lett..

[6]  Ioannis Chatzigiannakis,et al.  Distributed Circle Formation for Anonymous Oblivious Robots , 2004, WEA.

[7]  Franck Petit,et al.  Deterministic Leader Election in Anonymous Sensor Networks Without Common Coordinated System , 2007, OPODIS.

[8]  Nicola Santoro,et al.  Distributed coordination of a set of autonomous mobile robots , 2000, Proceedings of the IEEE Intelligent Vehicles Symposium 2000 (Cat. No.00TH8511).

[9]  Franck Petit,et al.  Circle formation of weak robots and Lyndon words , 2006, Inf. Process. Lett..

[10]  Franck Petit,et al.  Swing Words to Make Circle Formation Quiescent , 2007, SIROCCO.

[11]  Giuseppe Prencipe,et al.  CORDA : distributed coordination of a set of autonomous mobile robots , 2001 .

[12]  Branislav Katreniak Biangular Circle Formation by Asynchronous Mobile Robots , 2005, SIROCCO.

[13]  Franck Petit,et al.  Circle Formation of Weak Mobile Robots , 2006, SSS.