Optimal Control of Electrodynamic Tethers

Low thrust propulsion systems such as electrodynamic tethers offer a fuel-efficient means to maneuver satellites to new orbits, however they can only perform such maneuvers when they are continuously operated for a long time. Such long-term maneuvers occur over many orbits often rendering short time scale trajectory optimization methods ineffective. An approach to multi-revolution, long time scale optimal control of an electrodynamic tether is investigated for a tethered satellite system in Low Earth Orbit with atmospheric drag. Control is assumed to be periodic over several orbits since under the assumptions of a nearly circular orbit, periodic control yields the only solution that significantly contributes to secular changes in the orbital parameters. The optimal control problem is constructed in such a way as to maneuver the satellite to a new orbit while minimizing a cost function subject to the constraints of the time-averaged equations of motion by controlling current in the tether. To accurately capture the tether orbital dynamics, libration is modeled and controlled over long time scales in a similar manner to the orbital states. Libration is addressed in two parts; equilibrium and stability analysis, and control. Libration equations of motion are derived and analyzed to provide equilibrium and stability criteria that define the constraints of the design. A new libration mean square state is introduced and constrained to maintain libration within an acceptable envelope throughout a given maneuver. A multiple time scale approach is used to capture the effects of the Earth’s rotating tilted magnetic field. Optimal control solutions are achieved using a pseudospectral method to maneuver an electrodynamic tether to new orbits over long time scales while managing librational motion using only the current in the tether wire.