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What is different between Hill frame and local-vertical, local-horizontal (LVLH) frame in satellite? I add one figure to show this two frame.

enter image description here

reference of figure:

Weiss, Avishai, Uroš V. Kalabić, and Stefano Di Cairano. "Station keeping and momentum management of low-thrust satellites using MPC." Aerospace Science and Technology 76 (2018): 229-241.

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    $\begingroup$ I think your question is answered here but this is not my wheelhouse: space.stackexchange.com/a/32872/6944 See especially the comment by Julio on the answer. $\endgroup$ Dec 2 '21 at 14:34
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    $\begingroup$ @OrganicMarble This is my wheelhouse. One must step very carefully here because there is no standard definition of either the "Hill frame" or the "LVLH frame". One of the axes points along or against the vector from the center of mass of the central body to the spacecraft, another points along or against the angular velocity vector, and the third completes the coordinate system. Which of $\hat x$, $\hat y$, and $\hat z$ is radial, angular velocity, and completion varies; there are lots of choices. I've even seen the completion and ordering result in a left-handed coordinate system! $\endgroup$ Dec 2 '21 at 17:05
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    $\begingroup$ To the author: What you have depicted represents one concept of the Hill frame and one concept of the LVLH frame. There are other definitions; there is no standard. $\endgroup$ Dec 2 '21 at 17:12
  • $\begingroup$ slightly related with answers that don't directly answer this but may be potentially helpful: Are ECI and ECEF both frames and/or coordinate systems? Is there a difference? $\endgroup$
    – uhoh
    Dec 3 '21 at 1:18
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The terminology is unclear, so the correct answer varies depending on which book you read. For example, Curtis, Orbital Mechanics for Engineering Students, says they are the same, but Vallado, Fundamentals of Astrodynamics and Applications, negates the $x$ and $z$ axes of his LVLH. Neither of these matches the definition shown in the picture you posted, so at least for the purpose of reading that paper, follow its definitions, not anyone else's! Then, when you read the next paper, expect its choices to be somewhat different.

There are several reasons for this muddle. For example, in the term "LVLH", "vertical" is a vector, but "horizontal" is a plane. Therefore, any rotation of an LVLH system about its V axis has the same local horizontal, though the basis vectors chosen may differ widely. Also, when you're in space, what exactly does "vertical" mean? With respect to what?

Different authors make different choices. I described this in more detail here. David Hammen gave instructions here on various ways to make your own LVLH, with much the same warning about needing to adapt to the idiosyncrasies of whatever you're reading. The one Organic Marble's comment linked is very interesting because it shows yet another danger: the satellite in question is in orbit around the Sun, but the example LVLH frames are around the Earth. There are times when this sort of mixture of reference bodies is exactly the relationship you want for your frames, particularly if you need to adjust the attitude of an Earth-orbiting satellite in order to make its solar panels face the Sun, but it adds an additional layer of complexity that can hide nasty surprises.

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    $\begingroup$ I was going to write an answer, but this pretty much covers what I would have written. There is no standard that says "this is the Hill frame" and "that is the LVLH frame". In a way, Hill frame and LVLH frame are synonyms, but possibly with negations and reordering. One must be very careful when combining concepts from different sources. $\endgroup$ Dec 2 '21 at 16:58

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