Optimal Maintenance of Relative Circular Inertial Motion for Nulling Interferometry Applications

Spacecraft Formation Flying (FF) is a modern technology in Space Mission Design. Many recent missions have been designed based on this new architecture, in which a number of small, cost-effective satellites cooperate in space in order to achieve certain mission objectives. However, to ensure the feasibility of this paradigm, engineers have to deal with plenty of issues such as the necessity to maintain the formation geometry in a perturbed environment. A solution to this problem is given by optimal control theory, which allows to find the acceleration profile required to reach and keep the desired configuration in the most efficient way (e.g., by minimizing fuel consumption). Accordingly, a Linear Quadratic Tracker (LQT) and a Linear Quadratic Regulator (LQR) has been developed to study the formation keeping of two satellites in a J2-perturbed elliptic orbit. Despite a majority of investigations relies on the inaccurate Hill-Clohessy-Wiltshire model of relative motion, it will be shown that a more realistic dynamical model of the system dynamics leads to appreciable fuel savings. As a test case, a two-satellite configuration has been selected in which one spacecraft---the follower---orbits around the other---the leader---in a plane perpendicular to the line of sight to a chosen star, e.g. $\alpha$Cen, following a circular path covered in a given period. This is akin to the case of nulling interferometry on an Earth-bound orbit, an analogue of the back-up configuration of the L2-positioned Darwin mission. Here we study one of the several spokes or one collector-combiner pair of the full mission configuration. In particular, we first illustrate a Numerical Model (NM) of the relative motion that has been used in the simulations. This model is derived in the Earth Centered Inertial frame (ECI) and is valid for any eccentric orbit. Furthermore, it takes into account the second zonal harmonic of the Earth's gravitational field. Second, the optimal control profile for maintaining the configuration is found by solving the Differential Riccati Equation (DRE) either with the LQT or the LQR. Once the best control scheme is determined and the accelerations found, the $\Delta V$ required per orbit can be estimated and different configurations can be compared. Accordingly, the most efficient formations to satisfy the nulling interferometry requirements for satellites in Earth orbit can be determined. For example, we have investigated the difference in fuel consumption by varying the eccentricity and semimajor axis of the leader spacecraft. In addition, a parametric survey has been performed to quantify the effects of the target choice and the inclination of the leader's orbit on the total formation keeping cost. That is, the Delta_V required per orbit has been calculated as a function of right ascension and declination of the target and for six different inclination values.