Elsevier UK Chapter: 0GMA08 22-5-2006 9:22 p.m. Page:237 Trim:240mm×165mm Chapter 8 COOPERATIVE SPACECRAFT FORMATION FLYING: MODEL PREDICTIVE CONTROL WITH OPEN- AND CLOSED-LOOP ROBUSTNESS Louis Breger, Gokhan Inalhan, Michael Tillerson, and Jonathan P. How Aerospace Controls Laboratory, Massachusetts Institute of Technology Contents 8.1 Introduction  237 8.2 Dynamics of formation flight  239 8.3 Formation flight control and the model predictive control formulation  243 8.4 Distributed coordination through virtual center  249 8.5 Open loop robust control and replan frequency  260 8.6 Using closed-loop robust MPC  265 8.7 Conclusions  273 8.8 Nomenclature  274 References  274 8.1. Introduction Formation flying of multiple spacecraft is an enabling technology for many future space science missions including enhanced stellar optical interferometers and virtual platforms for Earth observations [1, 2]. Controlling a formation will require several considerations beyond those of a single spacecraft. Key among these is the increased emphasis on fuel savings for a fleet of vehicles because the spacecraft must typically be kept in an accurate formation for periods on the order of hours or days [3–5], and the performance of the formation should degrade gracefully as one or more of the spacecraft runs out of fuel [6]. This chapter presents a model predictive controller that is particularly well- suited to formation flying spacecraft because it explicitly minimizes fuel use, exploits the well-known orbital dynamics environment, and naturally incorporates constraints (e.g., thrust limits, error boxes). This controller is implemented using Linear Programming (LP) optimization, which can be solved very rapidly and has always-feasible formulations. The resulting algorithms can be solved in real-time to optimize fuel use and are sufficiently robust that they can be embedded within an autonomous control system. Efficient execution of precise formation flying relies on both accurate descriptions of the fleet dynamics and accurate knowledge of the relative states. Navigational errors [7, 8] and inaccurate physical models (such as ignored non-linearities, thruster misalignments, and differential disturbances such as J 2 and drag) can be significant sources of error [9]. 237