Fuel Efficiency of Fully and Underconstrained Coulomb Formations in Slightly Elliptic Reference Orbits

The Coulomb formation in slightly elliptic reference orbits for two different two-craft formation geometries is studied. The fully constrained formation requires both spacecraft to be aligned along the nadir direction. The underconstrained formation only requires the spacecraft to maintain a constant separation distance allowing each spacecraft to rotate freely on the surface of a sphere. The Udwadia–Kalaba (U–K) equations of constrained motion are applied along with a constraint stabilization method to obtain the minimum integrated control effort required for formation maintenance in both scenarios, where a logic is introduced to determine the maximum Coulomb force. In addition, two nonlinear optimal control strategies are applied for formation achievement and maintenance. For each formation geometry, the fuel efficiency as measured by the integrated control effort of the thrusters obtained using the U–K formulation is compared to that obtained using nonlinear optimal control strategies.