So let's review from last lecture. We went over classical Rodriguez parameters and we started modified Rodriguez parameters. So CK, what comes to mind with classical Rodriguez parameters >> [INAUDIBLE] from the quaternions. >> Do you remember? I'll just give you qi is equal to what? >> I don't know. >> Okay, is it Casey? Cody, I'm sorry. >> [COUGH] vi, probably not. So beta i, it's really the vectorial part [COUGH] of the quaternion scale by the scalar part of this. So Warda, what does this actually mean geometrically? I'm not going to go in order how I taught It. I'm going to try to mix it up, hope that things will stick a little better. Geometrically, how we are able to interpret this then? This isn't mapping, all right? We go from betas to qs, to CRPs. What's the geometric interpretation of this particular mapping? >> Mapping done. >> Okay, and we're mapping what, do you remember? >> Yeah, the surface, hyper-surface. >> Right? So this is my illustration of beta 0 squared, plus beta 1 squared, plus beta 2 squared, plus beta 3 squared equal to 1. So it's the unit hyperspherical surface, right? It's like just think of it like the Earth's globe. Earth is, pretend it's a sphere. This is it, and we're trying to peel that sphere surface off and map it onto a plane. And you know from maps that's never done easily. There's always distortions and issues and singularities that come with this. Good, so, Louis, what type of projection is this? >> Stereographic. >> Stereographic, so you need an origin and a plane. Warda's given you the plane, and I'm just writing it down. That was at beta 0 = +1. Where do I put the projection point? >> At 0. >> At the origin. So that's your projection point. So now if I have a point anywhere on here, let's just review this. If I put a point here, Casey, what type, Kaylee, why do I keep calling you Casey? Second person, you were Kyle, I called you Casey, so I apologize. >> [LAUGH] >> Anyway, what attitudes do I have here if I draw this point on this surface? Let's pretend this is beta 0 and beta 1, just one derotation. Keeps it easy. How far have you rotated if you're up there? >> 3 over 2. >> So how far have you rotated roughly? >> About 90 degrees. >> 90 degrees, right? I've drawn 45, but it's over 2. This is actually a 90 degree rotation. And is it positive or negative? >> Positive. >> Positive about your b 1 axis because this would be beta 1, right? That's how you can interpret that, right? So any point on this hypersurface represents an orientation. And of course the anti pull also represents, I'll just draw that one in green. All right, that's also there and you can project. Now the betas, do they map, non-dark, do they map to the same set of queues or do they map to this two distinct set of queues? because you got the one set that's the short rotation and the one that's the long. >> Same one. >> We had the same one. So with classic Rodrigues parameters, both of those map to the same set of queues. It's a one parameter set, good. To finish this, if you then have the projection point, you have a particular attitude, you connect the line between that beta attitude and this projection point. And that line has to intercept. And here it intercepts there, that's your qi coordinate. [COUGH] I ask you to derive this relationship, I think for the MRPs, I don't think I do it for the CRPs. But really, what you're looking at then is similar triangles, these kinds of triangles from here to here. because this coordinate is beta 0, this coordinate is beta i and that gets stretched to this distance is 1. You can create similar triangles. So that homework, once you see the geometry you can do it in two minutes. You're done. But until you see the geometry you might take you five, ten minutes to look at it. So that's what you want to look for there. So good, now it's a three parameter set. So Andrew, where do they go singular? >> 180? >> Yeah, you can everything except for upside down. If you go upside down, Maurice, what happens to these coordinates? How do they go singular? >> They blow up to infinity. >> Exactly, so those were the two types of singularities you encounter with three parameter sets. Where either there's an ambiguity like the 313, that inclination angle being 0, you'd have the 33 rotation which creates 0 over 0 ambiguity, some differential kinematic equations. Or as in this case, they will literally blow up to infinity. That could be bad, but it can also be a positive thing if you're doing feedback, if you have a situation where you just cannot afford to tumble. You can put a control in there, and if you look at your response function of the tangent curve, here at 180, your coordinates go to infinity. So if your control is -K times the set of coordinates, you're going to get, even with a constant gain, you're going to get increasing stiffness. And the response will get stronger and stronger to restore it. And you will see when we get to the control we'll be able to prove these types of things tend to be stabilizing even the non-linear sense. So it's nice. Some of these coordinates give you inherent behaviors that otherwise, if you do linear control, you have to do tedious gain scheduling. At this point, we'd use this case. Then we'd switch, and when we switch, do we switch smoothly? Do we increase it manually? All the stuff you have to do. By picking the right coordinate sets you can get such behaviors. So blowing up to infinity is not necessarily a bad thing. But if you're trying to model something, [INAUDIBLE] of a cube set, then it tumbles 16 times before you recover. Or for days, and then you have to recover with magnetic torque bars. This is not a good coordinate set to use. because you'll be running for singularities every tens of minutes, depending on your tumble rate, right? So pros and cons depends on what you have to do with it, good.