Formation Flying on Elliptic Orbits by Hamiltonian Structure-Preserving Control

摘要

The stabilization of relative trajectories about an elliptic reference orbit using only feedback from relative positions is investigated in this Note. Compared with previous controllers based on Melton's equation, the current controller derived from the Tschauner-Hempel equation consumes less fuel due to its Hamiltonian structure, which uses the center manifolds as the only feedback without the help of stable or unstable manifolds. In contrast to dissipative controllers, the proposed Hamiltonian structure-preserving (HSP) controller has an immediate effect of stabilizing the relative motion once activated. Its performances, such as the critical gain, controlled frequencies, foundational motions, and the maximum-likelihood optimizations are manifested in the (G-e) space. The semimajor axis has no effects on these performances except on the physical 3-D trajectories and fuel costs because it is eliminated from the formulations of the controller by length normalization. Two formation scenarios (a single formation consisting of a chief and a deputy and a double formation consisting of a superchief, subchief, and deputy) are discussed in this Note. It is also demonstrated that the performances of the double-formation controller can be evaluated by using only the eccentricity of the subchief no matter what the reference configuration is. The HSP control theory has applications in first- and second-order time-independent systems, as well as in time-periodic systems with single frequency. Future research could focus on time-periodic systems with double frequencies, such as the stabilization of formation flying in a J_2 -perturbed elliptic orbit, which have a fast frequency associated with the orbital period and a slow frequency associated with the evolution period of the mean argument of perigee.