Nearly fuel-optimal trajectories for vehicle swarms in open domains with strong background flows

Using a multi-vehicle control scheme based on smoothed particle hydrodynamics, a simulated swarm of unmanned underwater vehicles is guided along a pre-computed optimal trajectory between two points in an open domain under the influence of a strong background flow. The pre-computed trajectory is optimal in terms of fuel usage for a single vehicle. If the gradient of the velocity field is small compared to the total swarm radius, guiding a swarm of vehicles along this trajectory gives nearly optimal trajectories for all vehicles in the swarm without requiring additional costly optimization computations. We provide a bound on the maximum energy cost for vehicles in the swarm and also provide a more realistic estimate of the maximum energy cost. We also determine that the energy cost scales as N^3/2 for swarms with large numbers of vehicles, N. The algorithm and fuel cost bounds are verified in simulations of unmanned underwater vehicles moving across a double gyre system on the scale of a small ocean basin.