Statistical Analysis of Control Maneuvers in Unstable Orbital Environments

The optimal placement of statistical control maneuvers is analyzed for maintaining position near an unstable equilibrium point. This idea is developed for the libration points in the Hill three-body problem, but the analysis can be generalized to other unstable systems and is applied to the restricted three-body problem as well. First the basics of statistical fuel usage in the context of orbit determination errors and their mapping in time are reviewed. With use of linear theory, several explicit targeting formulas are devived for driving a spacecraft back to a fixed point. The mean and standard deviation of these schemes are analyzed for our special case, and explicit solutions for them are found. With use of these results the fuel-optimal spacing of the maneuvers in time can be devived to control a spacecraft to the vicinity of an unstable libration point. It is found that the optimal spacing is related to the characteristic time of the instability. It is also found that a linear quadratic regulator LQR control scheme will outperform control schemes that target the stable manifold of the equilibrium point.

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