Effective sensing regions and connectivity of agents undergoing periodic relative motions

Time-varying graphs are widely used to model communication and sensing in multi-agent systems such as mobile sensor networks and dynamic animal groups. Connectivity is often determined by the presence of neighbors in a sensing region defined by relative position and/or bearing. We present a method for calculating the effective sensing region that defines the connectivity between agents undergoing periodic relative motions. This method replaces time-varying calculations with time-invariant calculations which greatly simplifies studies of connectivity and convergence of consensus algorithms.We apply the technique to the case of agents moving in a common fixed direction with sinusoidal speed oscillations and fixed relative phases. For agents moving in a straight line, we show analytically how to select dynamics for fast convergence of consensus. Further numerical results suggest graph-level connectivity may be achieved with a sensing radius lower than that predicted by percolation theory for agents with fixed relative positions.