Optimal formation control for imaging and fuel usage with terminal imaging constraints

In this paper we build on previous work by the authors on spacecraft formation motion design for fuel-image optimality by Hussein, Scheeres and Hyland (2005). Previous results indicate a tradeoff between image quality and fuel expenditure by I. I. Hussein et al. (2005). In this paper we state results relating image quality to the trajectory of a sparse system of TV telescopes. This relationship is used to define a cost function for an optimal control problem including imaging and fuel performance measures. As in the work of I. I. Hussein et al. (2005), we derive the necessary optimality conditions for a generic multi-spacecraft formation and specialize them to a two-spacecraft formation. In this paper, image quality is imposed as a terminal constraint as opposed to being the object of minimization as in I. I. Hussein et al. (2005). We show that the optimal solution retains its symmetric properties and uniformity of the motion of its center of mass developed in I. I. Hussein et al. (2005). While the control law we derive here is much simpler than that found in Hussein, Scheeres and Hyland (2005), the formulation we propose results in a "speed control" effect that is desirable for uniform coverage of the frequency domain. Simulation results are provided to show the local behavior of the resulting control law. Finally, we prove optimality of a class of circular Earth-orbiting constellations introduced earlier in the literature. This work is fundamental to current and future work related to motion path planning of separated spacecraft interferometric missions used for the imaging of astronomical targets