# Locally optimal control for low thrust transfers with J2 and other Earth perturbations

## Background

As detailed in this answer low thrust optimization is subject of ongoing research. There are a few closed-loop control laws which allow to achieve a locally optimal (but close to globally optimal) transfer solutions between two orbits. Petropoulos, Naasz and Ruggiero have defined such control laws (and possibly others). The Ruggiero laws (1) are relatively simple in their implementation and have been shown to lead to almost identical results for fuel optimal and time optimal transfers the Petropoulos laws (2).

## Problem

Applying the locally optimal control laws of Ruggiero or Naasz for LEO transfers works like a charm in two body dynamics or with non periodic perturbations (such as Sun gravity). This has been shown in their respective papers and many have demonstrated this in software libraries (including in my own).

However, as soon as periodic dynamics are added to the propagated dynamics (specifically J2 in addition to two-body dynamics in as point-masses), the osculating orbital elements start to oscillate, causing these control laws to oscillate as well, and preventing them from converging in a reasonable time span.

How to use these control laws with higher fidelity dynamics such that they converge in a reasonable transfer time? I've tried computing the osculating orbital elements only at given spots in the orbit (true anomaly of zero, 45, 90, 270 degrees) but that doesn't help. Moreover, I'm concerned about using the Brouwer Mean orbital elements as these only smooth away the effects of J2: I assume that the control laws will start failing again once I enable spherical harmonics.

Follow-up question: Has anyone used the Eckstein Ustinov orbital element set developed in Spiridonova et al. 2014 for low thrust transfers?

## References

(1) Ruggiero, A., P. Pergola, S. Marcuccio, and M. Andrenucci. "Low-thrust maneuvers for the efficient correction of orbital elements." In 32nd International Electric Propulsion Conference, pp. 11-15. 2011. (2) Rabotin C. "Feasibility of Reusable Continuous Thrust Spacecraft for Cargo Resupply Missions to Mars" (2017). Aerospace Engineering Sciences Graduate Theses & Dissertations. link

• @uhoh, I edited to the question to fix typos and I hope clarify my problem. Let me know if further clarification or explanation is needed. Thanks! – ChrisR Jan 18 '18 at 5:00
• @uhoh, thanks for that paper. It seemed promising, but after further investigation by myself and another team member, we don't see how it actually changes the Ruggiero laws. The algorithm explained there is precisely the only way to implement the theoretical algorithms which work in two body dynamics, and how it's implemented in the two different software we use. I'm concerned we might have to rederive the control laws from another set of orbital elements, which will be tedious (but might be publication worthy). To be continued! – ChrisR Jan 19 '18 at 0:58
• I think you're on the right track with using mean elements for control, and I'd really like to see an answer laying out exactly how. – Chris Jan 19 '18 at 2:15
• @uhoh, sorry I didn't understand that point from your previous comment. I've added a background section which I can expand if you think the users of Space@SE might like. – ChrisR Jan 19 '18 at 5:27
• @Chris, I haven't yet tried to use the Brouwer Mean element set, but I'm concerned that they'll only smooth away the J2 effects and the other periodical effects will cause large transfer times. That being said, J2 is the greatest periodic effect, if I recall correctly, so it might work. From your experience, do you expect the Brouwer set to be sufficient to behave correctly in highly fidelity dynamics? Thanks – ChrisR Jan 19 '18 at 5:29