Control effort evaluation for low-altitude formation flying

Formation flying faces an increasing interest as a mission concept offering reliability and performances improvements. However, orbital control effort required to maintain the desired configuration can result too expensive, and therefore its evaluation becomes a critical point in judging the feasibility of a mission. A correct estimate will depend on the closeness to the real environment of the dynamic model used to design the control. In such a perspective, this paper introduces the Carter-Humi modification to the classical Euler-Hill set of equations in order to take into account the effect of the atmospheric drag. Low altitude orbits can be therefore modeled in a far more accurate way than using Euler-Hill schemes, while a simple model structure (linearity of equations) is still held. Based on this set, an optimal controller has been codified on the basis of LQR approach, detailing all transformations needed to this aim. Two sample cases, i.e. a leader-follower formation and a circular projection formation in LEO, have been implemented as tests for the controller. Performances have been evaluated in the frame of a complete simulation including all remarkable perturbing effects. The advantage of the strategy computed on the Carter-Humi modified model is in the order of 10-15% of the control requirement with respect to classical Euler Hill case. In a more general view, the computation of the actuators' effort, obtained with a model closer to the real orbital envirornment, is deemed to be more meaningful than the cases already available in literature. Requirements derived in this way should therefore be quite useful while assessing mission feasibility. Interestingly, the same approach could be extended to other perturbations suitable to a meaningful linearization.