You're going to go through levels of stability definitions. Each one will be stronger and stronger. Lagrange stability is one of the weaker stabilities but it's a good, It's a stable system. Sometimes we were very happy with Lagrange. So, what does that actually mean? So, Lagrange stability...The motion X is said to be Lagrange stable or bounded as it, kind of, a corollary, relative to this reference trajectory, like, to follow this path that we want to do. If there exists a neighborhood, such that if you're in that neighborhood, once you enter this neighborhood at a certain time, you're going to remain in that neighborhood. A lot of math. Basically it stays bounded. That's what they're saying. Once you start out with your errors, great, but then I get to a point where I'm within 10 degrees, I'm guaranteed to stay within 10 degrees. Right? And that's Lagrange's stable system. The key is this bound, Delta, is not a function of initial conditions. It doesn't matter if I started out spinning like crazy or barely moving. All I can guarantee is you're going to stay within 10 degrees. Even if you have infinitesimal small departures, I cannot guarantee you're going to stay infinitesimal but I can guarantee you're gonna stay within 10 degrees. They're not going to grow to infinity. Right? That's the kind of argument that we have. So, this is basically boundedness that we're doing. And again, key Delta does not depend on initial conditions. Some simple examples of this that you can think of. If we're looking at-- So, if we're Lagrange stability... Let's look at a system here with a spring in a mass. Right? And this is basically an oscillator. Your equilibrium would be right up here, right? At the origin. And good. So, now we're looking at the stability of that but we're also including a disturbance. And the disturbance is simply gravity. So, if you think of a slinky or a spring, you know, attached from the ceiling, it's not just going to be stuck to the ceiling, it's going to deploy some because gravity pulls it down. And you'll see problems like this. This is basically our-- For attitude, this will become our unmodeled external torque. What if a have solar radiation pressure? What if you have outgassing? What if the thruster is leaking? There's some torque that, you know, how does that impact my control all of a sudden? And we often end up with boundedness. So, with gravity what happens is this system, pretend there's a little bit of damping in there, you know, when you let it go it's going to do this stuff and then it hangs at a certain distance. Where the spring force cancels, the gravitational force and that's where it is. That's the new equilibrium. But what we're studying is the equilibria of the original motion or references relative to up here. So, even if I have very small deflections, it's still going to deflect out. And let's just say it goes to one meter, an easy number, right? If I had one millimeter deflection at the beginning, it's going to do this and go there. If I have 50 kilometers deflections, it will bounce a lot. It's a magic spring, you know, it's a very long spring. It'll bounce a lot and eventually settle to one meter, right? So, regardless of initial conditions, I'm always going to settle somewhere within a one meter bound. That's what I could predict, and that's been called boundedness. Right? So, if you hear Lagrange stability, think boundedness, and the bound is independent of initial conditions.
