Periodic Relative Motion for Spacecraft Formation in Elliptic Orbit and Its Application to Formation Reconfiguration

Abstract

Spacecraft formation flying has been widely investigated due to increasing interests in designing clusters around a planet and its applications including rendezvous and maneuver. Multiple spacecraft in formation have many advantages such as cooperative mission execution, reduced cost, flexible configuration, and robustness. Nonlinear relative motion and periodicity condition in elliptic orbits are required to achieve the precise and efficient formation flying and reconfiguration. This dissertation presents periodic relative motion, formation pattern analysis, and spacecraft maneuvers for the formation reconfiguration in elliptic reference orbits.

The major achievement of this study is to present a general periodic condition which guarantees the bounded relative motion of spacecraft formation flying in elliptic orbits. The periodic condition of the circular orbit is easily obtained from the analytic solution of the Hill-Clohessy-Wiltshire equation. For the elliptic orbits, however, the periodic condition is restrictively described in near circular orbits and elliptic orbits at a specific initial position by the Tschauner-Hempel equation due to complex and coupled relative motion dynamics. In this dissertation, the general periodicity condition is derived using two analytic approaches: the first one based on the state transition matrix, and the second one based on the energy matching condition. Furthermore, the offset condition is investigated to make the leader spacecraft locate at the center of the formation geometry. As a result, the periodic relative motion is developed to remove the secular drift and the offset in the along-track direction. Numerical simulations verify that this periodic condition covers partial periodicity condition of previous studies. The periodic relative motion provides a substantial advantage in the sense that an additional correction is not required with respect to the initial position of the follower spacecraft.

The second accomplishment is to design the formation geometry and to analyze the formation pattern in the elliptic reference orbit. From understanding the natural formation geometry, the spacecraft can remarkably reduce the fuel consumption. The formation design method for circular or nearly circular orbits has been described

however, the formation design and analysis in the elliptic orbits have not been extensively studied due to the nonlinearity and eccentricity. In this dissertation, two formation geometries in the elliptic orbit are designed by considering natural periodic relative motion: radial/along-track plane formation and along-track/cross-track plane formation. The formation patterns including the relative trajectory, velocity, and formation radius are analyzed with respect to the variation of eccentricity. The eccentricity of the reference orbit is a critical factor influencing on the variation of formation radius between two spacecraft in the relative motion.

With the understanding of formation flying in elliptic orbits, the spacecraft maneuver problem for the formation reconfiguration is investigated considering two types of control input: continuous control input and impulsive control input. The formation configuration should be changeable according to the mission requirements and environments during the operation. The follower spacecraft changes the formation radius or the formation geometry with respect to the leader spacecraft, while minimizing the control effort. For the continuous control input, the optimal control problem in the relative motion is solved by the Gauss pseudospectral method, where the initial and final relative positions and velocities are specified. The optimal trajectories between the radial/along-track plane and the along-track/cross-track plane are provided using the minimum energy and minimum fuel cost functions. For the impulsive control input, the Lambert's problem is modified to construct the transfer problem in the relative motion, given two position vectors at the initial and final time and the flight time. In addition, the minimum velocity change for transferring orbits is designed through the grid search.

The results of this research establish that the periodic relative motion is presented at an arbitrary true anomaly in the elliptic reference orbit, the formation geometries are designed and analyzed to reflect the eccentricity of the reference orbit, and the formation reconfiguration is described using two control input types. These results can provide the effective and efficient formation flying in elliptic reference orbits.