Moving on, we're leaving all Euler angles behind. They're a great building block. They're very useful. It's fundamental thing. But for general spacecraft motion I don't tend to use Euler angles. I may use them for sub components, so other kinds. I find places where they're very, very useful. But for the general 3D tumble they're not good because we're never further than a 90 degree rotation away from a singularity. Now we're going to look at the principal rotation vector. That's what we call it typically in astro. Well principle rotation parameters, you can see one's a three dimensional set, one's a four dimensional set but they are very, very closely related. If you do robotics, I know Sandia has some called the Rodriguez parameters. Which is really confusing because we have also Rodriguez parameters in something very different. So if you're reading papers and different literatures find the math. The math tells me precisely what these coordinates are. It's just labeling. So they're very important and we'll be using this concept over and over again. And it all comes from Euler. [SOUND] Euler was a mathematician with way too much time on his hands. Can you think of a scientific field that doesn't have an Euler's theorem. And if you talk to somebody about Euler's theorem in mathematics versus in aerospace, versus in nuclear physics, or who knows. Everybody has Euler's theorem. I mean this one. No, I mean this one. Amazing mathematician. This is one of them. This is my Euler's Principal Rotation Theorem, gotcha. Says a rigid body where a recorded reference frame can be brought from an arbitrary initial orientation to an arbitrary final orientation. By a single rigid rotation through a principal anglen phi, and a principal axis, e hat. The principal axis is a judicious axis fixed in both initial and final orientation. All right, what on Earth, who talks like this? What does this actually mean? So we're going to go through some of these copies and details because these other this parameterizations, all exploit nuances of these properties. And if you understand this well the next week of three lectures are going to be that much more obvious. It won't be, whoa, where did this one come from? It's like no, it all comes from here, okay? So we're going to spend the rest of the class to do this a little more detail. What it essentially means is you can go from anywhere to anywhere, attitude wise. There's no singularities. That would be an easy way to say, but no they're kind of at a loss for words. They have time on their hands apparently, this mathematicians. All right. But essentially from any attitude to any attitude that's what the first part means. Second part by means of a single rigid rotation. So if you think of all other angles, we go from anywhere to anywhere. But by means of three sequential rotations. So he got pitch row, and you can have a yaw a little bit of a pitch and then a row that almost pitches you back. You can do this stuff with big angles where you're almost back at the same orientation. We've seen examples of that. Two big order angles and the difference is kind of small. So this takes you from anywhere to anywhere using only one angle. Se we're using this capital Phi letter for that principle rotation angle. And so I go from here, and then I have some axis. There we go. This will work. So you've got a frame. You can put some axis. Compared to other angles, we had to always rotate about one, two or three. That was it, right? The base vectors. And that would give you the sequence. Here we will have this e hat vector, which is this axis we have to figure out that says, look you stick this axis on here. And then we twist about that axis by the angle phi and I get to the final orientation. So instead of a three sequence about only base vectors, I'm allowing a general univector about which I can rotate. But I can always get from one frame to another with one angle about that axis at that instant. That's what this theorem says. This is the image of it. So here if we do this you can see the endframe in grey. The b frame. And I'm rotating so the dashed line is an extension of e hats and somehow just take a frame. You stick this knitting needle in there, something long. And then you can twist about it and that's how you're rotating. So this is where we get there. The other thing is the last part is to kind of an oddly stated thing. The principal axis is judicious axis fixed in both the inertial and the final orientation. So fix in both orientation means that vector appears exactly the same in one frame and in the final frame. It'll have the same vector components in b and n frame. So, and it's a little bit fuse, see the image you can kind of imagine that, okay, I can see roughly how that might be happening geometrically. But we'll look at all the mathematical implications of this theorem. So if I have Euler angles it's a three sequence, three, two, one, 60, 50, ten and you get there after a sequence. Here I'm doing the principle rotation vector I've drawn in red. That's the e hat that I had to figure out. And I rotate there. So I'm rotating only about one axis in place, and at the end I have to rotate 80 degrees to get there. Now here immediately I see a benefit compared to Euler angles. If somebody tells me, look the attitude difference between b and n is an 80 degree rotation, I know right away they're not close. That's a tracking problem. I'm doing a horrible job with my control if that's what I wind up with, right? I'd want to be much closer. If somebody gives me all these other angles I don't know if that's close or not. Especially with big angles I have to go through the math and see what's the actual angle of difference. So this set is much more convenient in that sense. You can quickly say, well that's about a three degree rotation about some axis or that's a hundred and twenty degree rotation. That's really far, right? So we can get there and that's the unit axis that you'd have. So cool, so this will be now our principal rotation parameters. If I talk about the parameters, it's really the four dimensional set. It's the three vector components of ehat and this principal rotation angle. So we've moved from three parameters to four parameters. Jordan, how many constraints do I have. >> One. >> You're my constraints expert. One, right. Can you tell me, Kevin. All right. Can you tell me where that constraint must manifest? Somehow these coordinates, four coordinates must be constrained. Where is that? All right, these are my principle rotation parameters here. CK what do you think? >> The e hat has embedded. >> Exactly, it's all embedded in here. You can't have one, two, three. That is an axis, yes. But all this math will depend on e hat being a unit direction vector. So we're back down. Once you know two of them the third one is constrained to make it unit length. So you still end up with a three degree attitude problem but have moved to four coordinates. So now many of you have actually seen these. Okay, there's a question in the back. >> Yeah, why does it matter if it's a unit length vector because it's just a line vector of the direction cosine matrix. That's how you'd find it. So it why would the length actually matter? >> The way it's typically defined is this way. If it's not a unit vector, then you would have, well you're jumping ahead. We'll get to that part of property in a moment. [CROSSTALK] >> Okay. >> Do you mind if I just defer that? >> Sure. >> When I talk about that, then this will make more sense. So if this is a four dimensional set, is this a non-singular set? We said if it's three dimension. And if you have three parameters, there will be a singularity. So if it's a singularity. If you have a singularity, think of ambiguities. Is there rotation? Where I can't go, I don't know what the final intension is. Or I can't go backwards that's what happen with three, one, three, so in connection of three rotations, I can do one way the math. But if I know the total rotation at 60 degrees about three, I can figure out which two angles got you there. So yes. >> Rotate at zero or all the way around 180. Then you wouldn't know if it was backwards. >> Exactly. The zero, one is the easiest to visualize. If I have a zero rotation angle, great. But how do I, about which axis do I not rotate and stay in the same place. That's essentially what you're asking. And of course the answer is any. You could rotate about not any axis and not have moved. So that's perfectly fine. So yes. So this is an illustration that just moving beyond three coordinates does not guarantee you you're going to be nonsingular. In fact principal rotation parameters is a classic problem where there not, they are, there have gone before and we have singularities. Why are we doing this? You will see there a lot of benefits with this too. So there are a singular set and you mention zero, 180, 360, all correct. All will come back in place.