Fuel-/Time-Optimal Relative Trajectories for a Satellite near a Perturbed, Elliptical Orbit

The problem of fuel/time trajectory optimization is considered for a spacecraft moving relative to another in a perturbed, elliptical orbit. The relative motion model for the deputy satellite with respect to the chief is assumed to be well-approximated using the Gim–Alfriend state transition matrix. Control is applied in the in- and cross-track directions, which makes this an underactuated but reachable system for any eccentricity and inclination (excluding the critical inclination). The system is discretized using a zero-order hold on the input, and the control signals are computed using a linear program, which results in a bang-off-bang control profile. A balance between the time of flight and the required fuel are analyzed using a genetic algorithm and the results are compared with two well-known maneuvers from impulsive orbit theory. The results of the new method agree well with the impulsive solution in terms of the velocity change required.