Optimum Guidance Laws for Low-Thrust Orbital Maneuvers Using Equinoctial Elements

The use of equinoctial elements and their corresponding variational equations alleviates the problems of singularity and instability that are associated with the classical orbital elements. As such, the equinoctial elements are well suited for the long-term simulations of satellite orbit. Compared to the classical orbital elements, however, the equinoctial elements are relatively abstract as they provide little direct insight into the shape and orientation of an orbit. The motivation of this work is to derive optimum guidance laws in terms of the equinoctial elements, for the adjustment of each of the classical orbital elements. The optimum thrust laws are obtained by analyzing Gauss' variational equations through the inclusion of two steering angles in their formulations, namely azimuth (in-plane) and elevation (out-of-plane) angles. In particular, considering the classical two-body problem with the inclusion of the thrust force as the source of perturbations, the optimum thrusting strategies are derived for the modification of each orbital element. A number of case studies are presented, in which one or more elements are sought to be adjusted. Thruster characteristics and the amount of consumed propellant in each maneuver are determined using the performance parameters of a commercial-off-the-shelf, state-of-the-art low-thrust propulsion system intended for SmallSats. For the scenarios where more than one element is to be modified, the Directional Adaptive Guidance law is utilized which synthesizes multiple thrust directions into a single thrust force for inclusion in the variational equations. A detailed discussion on the tuning of the weighting factor and the adaptation law that are included in the Directional Adaptive Guidance law is presented, and some strategies for expediting convergence to the desired orbits are discussed. The results show that, for the long-term simulation of spacecraft's trajectory, the equinoctial-based, optimum guidance laws are most suitable, and they can handle the situations wherein classical orbital elements and their variational equations behave erratically due to the singularity of one or more of the elements. As illustrated in this paper, one such scenario is when the spacecraft is to be transferred to the geostationary orbit, where orbital eccentricity and inclination are zero, rendering several of the classical orbital elements undefined and Gauss variational equations unstable when the spacecraft nears the target geostationary orbit.