Trajectory optimization for multi-agent persistent monitoring in two-dimensional spaces

We address the persistent monitoring problem in two-dimensional mission spaces, where the objective is to control the movement of multiple cooperating agents to minimize an uncertainty metric. In a one-dimensional mission space, it has been shown that the optimal solution is for each agent to move at maximal speed and switch direction at specific points, possibly waiting some time at each such point before switching. In a two-dimensional mission space, such simple solutions can no longer be derived. We approach the problem by representing an agent trajectory in terms of general function families characterized by parameters that we can optimize. We then show that the problem of determining optimal parameters for these trajectories can be solved using Infinitesimal Perturbation Analysis (IPA) to determine gradients of the objective function with respect to these parameters evaluated on line so as to adjust them through a standard gradient-based algorithm. We have applied this approach to the family of Lissajous functions as well as a Fourier series representation of an agent trajectory. Numerical examples indicate that this scalable approach provides solutions that are near-optimal relative to those obtained through a computationally intensive two point boundary value problem solver.