# What is different between Hill frame and LVLH frame in satellite?

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

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.

• 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. Dec 2, 2021 at 14:34
• @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! Dec 2, 2021 at 17:05
• 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. Dec 2, 2021 at 17:12
• 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?
– uhoh
Dec 3, 2021 at 1:18

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.