What I want to do next is do a review of a principle rotation vector, because a lot of the things we do today, we will do Thursday. And I think next Tuesday we're wrapping up all the kinematics. So we're going to start to go through these problems quicker and quicker and quicker. The basic schematics of how do I add, how do I subtract, all those kind of things I'm expecting to stick more and more and more, right? So you do need to become more and more comfortable a little bit what these things mean. And then we're looking at the differences between them. Where does this one behave differently? Why is it good and what are the behaviors? That's kind of what we'll be focusing on more and more. So let's review these things. If I have e hat and fee. Those are what I would call principle rotation parameters. It's a four parameter set. Is this four parameter set singular or non-singular? >> It's singular. >> It is singular. So even though we're going to four parameters, kind of like quaternions, it doesn't guarantee you have a non-singular set. Okay, good Marty, where are singularities with these sets? >> When we have an ambiguity in the direction that were rotated about the vector. >> Yeah, so for which principal angles do we go singular then? >> The first? >> There's only one angle, so yes, it's the first. [LAUGH] >> [LAUGH] >> Right? >> I forget the name of it. >> The principal rotation angle. So for which principal rotation angles do we go singular? When is it ambiguous? Anybody. >> Zero. >> Zero's one, and what was the other one actually? >> 180. >> 180's another one. Not quite as easy to illustrate, but there's many ways you can flip upside down about different axes and come up with the same orientation. So then there's all of a sudden an infinity of answers. And that's what leads to this mathematical issue. So I'm going to solve that in a differential kinetic equation. So we had 0 over 0 types of ambiguities in there. Okay, good, so they are singular. Why do we like them? What are some benefits? What can you tell me? Is it? What are some good things about these? >> Easier to visualize. >> Yes, true, they are easier to visualize. Let me see, okay, it's working again. Good, but and why are they easier to visualize? >> [INAUDIBLE] >> Well, this is the key thing really, it's this angle. We often, instead of doing a 3-1-3, you could actually do a large angle, small inclination, and then minus a large angle, almost back at the same place. But in other angles, you're looking at two large ones, and it's not always obvious if there's a small or a big difference. And especially later next week we'll be getting into estimation stuff too. Whenever we have to say how good is our estimate. This is the actually true body frame and then we're estimating one that's a little bit off. And if given in terms of three one threes, it might go, wow, that's horrible. But turns out, no, you're only four degrees off. So this is very nice, because it gives you a single one angle. And then we're doing an estimation also, I just care that I am in within one degree. If you're doing pointing, typically we don't care if we're up or down off or left or right, there's no biasing in that. I just care that I am within one degree. And this gives you a single angle that you can use. So that's a really cool thing. So a lot of students or researchers actually use this when we do plotting or performance. I may have done my stuff using quaternions, but what I show in the end is, hey, my principle angle got within one degrees of where I wanted it to be. I don't care about which axis I'm off, right? So that's a very handy thing. The other thing is we do get there in a single axis rotation, which is sometimes handy even for large rotations. If I'm 170 degrees about b2, or mostly about B2, I can quickly visualize in my head, okay, that means I am that far down. So, that's kind of a nice thing too. Good, how many, so Robert, are these [INAUDIBLE] parameters, are they unique? For an attitude, is there only one set of parameters that defines this orientation relative to this orientation? [INAUDIBLE] two because you're not. >> Let's just say my one, two and three axes comes out of the paper. I've rotated 45 degrees in a positive way about three. Is there only one way I can go from this orientation to a 45-degree rotation? >> You can go the other way. >> Okay, so another approach is you could go this way, in which case, what was that, 315, I think, if I did my math right in my head, minus 315. That's there. Now, we have two options, Tebow, is that it? >> You could, I mean, it's kind of the same thing but you could say negative e and then rotate positively of 315 around negative e. >> Right, so what Robert was talking about is about a rotation, there's always two options. There's a short one and a long one. And at 180, short and long are the same. Yes, right. But generally speaking, there's always a short rotation and a long rotation, right? It's not like all the angles are all of sudden, hey I'm doing a 3 and a 3 rotation, and there's an infinity of options. It's really just two, that's it, if we have one axis. But as Tebow was saying, you can actually flip that axis. Instead of doing a positive 45, I can also consider the down axis. And in that case, it's a -45, right. And then of course, it's also again a short and a long. So we end up with 4 sets of principal rotation parameters. So from the DCM, when we did that math, there are actually four different ways that you can get. There's always a short angle and a long angle and the e hat has a sine ambiguity that you can do and then the angles resolve themselves around that. So there's four different ways we can do that. Okay, good. Any questions on principal rotation parameters? We have the principle quotation vector. We should do a little bit in the homework. I typically write just a gamma as fiat times niat. Alright that's the three parameter set. Has the same similarities. Zero, 180, and we saw the differential kinetic equations. We won't use those much in this class because really, from a control application, why would I use coordinates that go crazy as I get to my desired state? That's just asking for issues, right? So good. Principle rotation parameters.