Let's start out with a general review again. So thanks for showing up in the audience today. Let's go through what we did this morning actually which was actually pulling in together different structures. So Trevor if we're going to do kinetic energy, what was a good positive definite function we could use for kinetic energy? Kinetic energy, rate, I gave you the answer. Okay, kinetic energy. So we used kinetic energy as a good measure if we're only doing rate feedback, which is one 1/2omega(i) this, right? That was a nice prototype. If we want to do position errors, what kind of function can we use? >> The same kind of function [INAUDIBLE]. >> Okay so let's say if we use Euler angles. You can do various similar stuff, it's just Euler angles transposed with some positive definite gain k, that you can throw in. Definitely submit, this is a positive definite function that we need, good? And the different prototype functions. Anybody remember what we did with Rodriguez parameters? Instead of the natural log. That was a nice trick though. That's not a mechanical analogue. If somebody looked at the math carefully, came up with these things and found wow, if I use, let's do CRPs. And then we use the natural log. I forget if there's a factor of two in here or not. But if you use something like this. And then take its derivative, u maybe does a gain in front of it. Then v dot ends up being minus k omega transposed q, right? And your altitude actually appears linearly, which is great then from a control perspective. And the same thing with MRPs and we also looked at. So we now have kind of basic building blocks. The things to remember here, if we're doing just regulation you can typically write your angular velocity relative to inertial, this is true kinetic energy. If we're doing tracking then delta omega ends up being omega B relative to N, minus R relative to N, which is B relative to R. And that means here, well, for the rate tracking, it made the control more complicated. For the additive Lyapunov function, we still used exact same form. You just say, hey, now Euler angles to find altitude of B relative to R, instead of B relative to N. And then the kinematics work themselves out, so it's a purely kinematic result. So regulation tracking, here, was pretty much the same. For the rate part, we throw in these deltas here, and give the same mathematical structure. So good.