Optimal Control of Active Drifter Systems

Drifters are energy-efficient sampling platforms for waterways and other water bodies with pronounced flows. The motion of passive drifters is determined by the underlying flow and thus limited. To overcome this limitation and enhance maneuverability, we consider an active drifter, which has a variable control surface for modulating the hydrodynamic drag force, and a thruster for propulsion or braking. In order to maintain the active drifter as an energy-efficient platform, the use of the thruster must be sparing and carefully considered. In this paper we present an optimal control problem for a one-degree-of-freedom active drifter system, where the cost function aims to balance the objectives of shortest time and minimal thruster use. Despite the complex, realistic nonlinear dynamics of the active drifter, an analytical solution to the optimal control is found by exploiting PMP. In particular, for a given desired final state, each candidate optimal control solution is propagated backward in time, to compute the points in the state space where the optimal control switches; such points are parameterized by the co-state variables, linked implicitly to the system's initial conditions. The proposed approach naturally results in a final-state-dependent partition of the state space, where each region corresponds to a given optimal control value. A feedback control law is readily derived from the aforementioned map. The efficacy of the proposed approach is supported with a numerical example, where the trade-off between time-optimality and fuel-efficiency is illustrated.