Are LVLH and RSW coordinate systems the same thing?

I've been reading into satellite relative motion, and I noticed that the literature split on using LVLH (local vertical local horizontal) and RIC/RSW (radial intrack crosstrack) coordinate frames. When I've looked into the definitions of each, it appears that LVLH has a z axis that points to the center of the earth, a y axis that is negative to the orbit normal. In contrast the x axis of RIC points in the direction of the spacecrafts position vector, and the z axis is pointing in the direction normal to the orbit plane.

However, I've found other sources, that have said RIC and LVLH are the same thing. Could anyone clear this up?

EDIT: Here is the source: https://public.ccsds.org/Pubs/500x0g4.pdf on page 23 (section 4.3.7.2) the LVLH frame is defined. In section 4.3.7.4 the RSW frame is defined, and the author says that RIC and RSW are the same thing. Should I just stick with Vallado's definition because his work seems to be the preeminent book in the field?

• Partial answer: the LVLH (Local Vertical Local Horizontal) name often is only enough information to know that two of your axes are aligned that way. The ordering and sign of the axes often varies depending on the convention of the algorithms you use. For example, the CW equations have Y radial, Z along angular momentum vector, and X as the cross product of those (velocity vector when circular). Nov 16, 2020 at 2:11
• @uhoh There you go. Nov 16, 2020 at 9:36
• You can use the Contact Us form to have your accounts merged, so that you can freely edit your question. Nov 16, 2020 at 9:41
• @uhoh fixed it! thank you
– Syd
Nov 16, 2020 at 13:10
• @Syd Thank you for sharing the CCSDS standard. I also want to give another source pointing to the same conclusion as your reference. The Space-Track.org Handbook for Spacecraft Operators has a footnote (page 41), equating the RIC and RTN frames. space-track.org/documents/…, I have been skeptical about this "simplification", and have been trying to be a bit more pedantic, because the book's explanation is more complete and has a rational for such a distinction. Nov 18, 2020 at 1:41

1 Answer

There are many ways for definitions using the same names to disagree. Any attempt to be complete must be inconsistent, and any attempt to be consistent must be incomplete, because each word in the language of the question has historically been translated into and from mathematics in multiple ways. The most honest answer we can give is that it depends on the choices made by the authors of the document you are reading. In some books they coincide, but in others they don't. Even different editions of the same book by the same person can disagree, because in the years that passed between two revisions, the writer may have changed their mind. The best anyone can do is try to be unambiguous in defining each term they are going to use in the rest of the document they are writing, with some examples of how to translate that into the different meanings for the same terms that have been used by some other documents. This is true on the ground as well as in space --- when you say "local vertical", do you mean perpendicular to the reference ellipsoid, perpendicular to the geoid, perpendicular to the terrain, or something else?

Not all authors restrict themselves to right-handed, orthonormal frames. At the level of pure mathematics, all you need is a three-dimensional vector space at every point in three dimensions, but you can use a different vector space at every point, and the choice of connection between the vector spaces used at different points is arbitrary (and the term connection has a specific technical meaning which differs substantially from its use in other subjects). I would argue that the meaning most often encountered in spoken language is one of these: rather than argue about which one of "in-track" and "along-track" is parallel to velocity and which perpendicular to radius, most people talk as if they use unnormalized $$\vec{r}$$, $$\vec{v}$$, and $$\vec{h}=\vec{r}\times\vec{v}$$ (angular momentum per unit mass) as the local basis vectors, and just live with the fact that the angle between $$\vec{r}$$ and $$\vec{v}$$ is variable. I'll also note in passing that in my head, $$\vec{r}$$ is perifocal, but not everyone will think that, which has a significant impact on the definition of these frames for planets (orbiting the solar system barycenter, not the center of the sun).

Even for orthonormal frames, there remains the question of which direction gets which label. Where does the spacecraft +Z axis point? Anywhere the person talking about it wants, as long as they say what it is. There are no enforcement mechanisms, except possibly at the level of "if you work in my company you will use definition Q and no other or I will fire you"; but even if every satellite builder tried to do that internally, they would still end up picking different ones than their neighbors, causing havoc when trying to collaborate. A paying customer can incorporate any document they want into a contract, but each contract could be different. The standards making bodies recognize this, so they only make recommendations. For example, ANSI/AIAA S-131-2010, the American National Standard for Astrodynamics — Propagation Specifications, Technical Definitions, and Recommended Practices, closely resembles Vallado's book because Vallado was on the committee that wrote it, but all it does is list a bunch of ways it could be done, and then says "The use depends on the particular organization. Consistency and documentation are the important requirements."

• Thank you for sharing the AIAA standard! Your last point on consistency and documentaion nailed it! From my end, I tried to point to a broadly adopted source, but the truth is exactly as you describe it. Dec 3, 2020 at 19:38