Design and Control of Libration Point Spacecraft Formations

Abstract

We investigate the concurrent problem of orbit design and formation control around a libration point. Concurrency implies that the design and control problem are simultaneously investigated. Separating the two problems is both unnecessary and ill-advised. The full problem can be naturally cast as a multi-agent, nonlinear, constrained optimal control problem. The optimality criterion is fuel consumption because the engineering feasibility of a formation design is dominated by the amount of propellant required to maintain a formation. Contrary to popular belief, quadratic costs do not measure fuel consumption; consequently, we take a direct measure of fuel consumption given by the L1 norm of the control acceleration. Fuel budgets to individual spacecraft are allocated by isoperimetric constraints. As with most nonlinear problems, the resulting problem does not have closed-form solutions. The full problem is solved by a Legendre pseudospectral method implemented in DIDO. DIDO exploits SNOPT, an active-set sequential quadratic programming solver, and generates quick solutions to facilitate redesign, an important requirement during the early stages of formation design. This approach does not use linearizations in modeling the dynamics, nor does it require analytical results; rather, the inherent nonlinearities associated with the problem are automatically exploited. Furthermore, we take advantage of a true distributed system architecture that does not rely on designing a leader–follower system. Sample results for formations about the sun–Earth and Earth– moon L2 point in the three-body circular restricted dynamical framework are presented. Optimal solutions for relaxed and almost periodic formations are presented for both a large separation constraint (about a third to half of orbit size), and a small separation constraint (about a millionth of orbit size).