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Suppose we have our spacecraft in a circular orbit around the Moon and we want to transfer it to an elliptic orbit with a medium-thrust (10-100N) retrograde burn. Based on the current state estimate, we want to determine a propellant-optimal steering strategy to achieve the transfer. The thrusters operate at full throttle when they're activated, and thrust levels decline with propellant consumption (the propulsion system operates in blow-down mode).

It seems to me that it is best to consider such a problem as a two-point boundary value problem, with the initial and final state vector given by some set of non-singular orbital elements (such as the equinoctial elements), and the dynamics given by some accordingly modified form of the Gauss planetary equations. Are there any standard and/or proven methods to solve this problem in real-time, aboard the spacecraft? Or are there other common guidance strategies that are better recommended?

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The requirement of solving this particular guidance problem in real-time on-board the spacecraft is really exigent. As far as I know the usual thing to do is solve the guidance problem on Earth and then uplink it to the spacecraft. Maybe the spacecraft control, understanding control as the ability of the spacecraft to follow the desired path in the presence of disturbances, can be autonomous by some sort of feedback law, MPC or event control. But maybe not because the dynamics are so slow and it may be more simple to manage control from Earth.

Anyway when facing such problems I barely always find direct methods much easier and intuitive to employ rather than the indirect TBVP's. The key idea of direct methods is first discretize and then optimize whereas indirect ones do in opposite order. A good book is "Spacecraft Trajectory Optimization" written by Bruce A.Conway.

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  • $\begingroup$ Thanks for your response. In Falck et al. (2014) (arc.aiaa.org/doi/10.2514/6.2014-3714), two closed-loop guidance algorithms for low-thrust vehicles are described and compared. Can't these be used to solve the guidance problem in real-time? Or do I understand their working incorrectly? $\endgroup$
    – woeterb
    Oct 15 '18 at 9:29
  • $\begingroup$ Taking a glance at this paper, what I see is that it can be implemented in real-time because it employs sub-optimal laws with analytical expressions. If you can find a form to solve your problem analytically, or at most requiring an optimizer to solve a LP or QP with a moderate number of parameters, then ok. $\endgroup$
    – Julio
    Oct 15 '18 at 12:43

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