In Chapter 5 presented and analyzed the SIS Algorithm that synchronizes a col-lection of n agents or beads, moving on a ring, so that each bead patrols only a sector of the ring. The algorithm is distributed and requires that two agents exchange information only when they meet. We have established that the proposed algorithm renders locally attractive the periodic modes corresponding to balanced and unbalanced synchrony. Sim-ulations indicate that convergence to the desired periodic modes takes places for a large set of initial conditions.
Without providing a formal analysis, we mention here a few properties of the pro-posed algorithm. The SIS Algorithm (1) adapts smoothly to arrival and departures of agents throughout execution time, including adapting to switches between odd and even numbers of agents, (2) handles smoothly measurement noise and control disturbances, (3) has memory requirements and message sizes independent of n, (4) is truly distributed and does not require agents to have unique identifiers, and (5) is invariant under rotations and reflections.
Furthermore, our algorithm may be implemented even on robotic agents that do not have access to their position with respect to a global reference frame on the ring, i.e., even if they do not agree upon the position of the absolute 0 angle. To be specific, assume that each agent can only measure the angular distances that it travels and that,
the impact position. Then, it is easy to see that this “relative angle” information suffices to implement the update rules of the feedback law.
One may design alternative approaches to the basic problem of steering a group of agents to balanced synchronization. An example alternative solution is described as fol-lows: all agents could rendezvous at a common location, thereby forming a connected communication network; then they could elect a leader, agree upon an open-loop plan, and implement it without any further communication. This approach is philosophically and practically very different from our proposed algorithm. We leave a detailed compar-ison to future works.
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