So now we're going to talk about Modified Rodrigues Parameters. These are quite popular, used in lots fo different areas. Having just gone through Rodrigues you will see that we can go pretty quickly here. There's one subtle difference, but man, this little difference makes a huge impact on how we can use them. Instead of defining, like CRPs we had Beta I over Beta Nought. Now, we have Beta I over 1 + Beta Nought. So Warda, she was saying earlier that well these are singular if Beta Nought went to 0. If I have a 0 rotation Beta Nought is 1, so I divide it by 2, easy. If I have upside down, Beta Nought goes to 0. I still have 1, that's not bad. So for what orientation now, will these go singular? Go ahead. [COUGH] >> When it's upside down, 180? >> No, 180 Beta Nought is 0. >> So then 360, when it's- >> Yeah. >> At 1. >> Okay, so wait a minute. So if I'm telling you, this is the inertial frame, this is the body frame, and they're identical, do you have a 0 or a 360 rotation? So is the MRP singular or non-singular? >> If you've done nothing, it would be zero, right? Until you start cycling through. >> Right, and how did you know it did nothing? All you know was they’re identical. So at this instant is my MRP description is singular or non-singular? >> SIngular, because you don't know? >> Who thinks non singular? Who thinks singular? Who is confused? >> [LAUGH] >> Honest people. Good! I hope you're a little bit confused. That probably means you're thinking about this and go, wait a minute! What's going on here? This is where it starts to get a little bit weird. But man it is great, it is really cool how we can take advantage of these things. So, as you're very correctly summarizing, you have to know the path, and if they're just identical right now, you get to choose did it do one revolution or not, right? And I typically say no, because who wants to deal with singularities, right? So the singularity has been moved from 180 degrees all the way back to 360 and 360 to same as the origin. So that should give you a little bit of a headache until you kind of go wait a minute. Okay this starts to make sense. So now the MRPs are like CRPs. They're a ratio of the quaternion vector part over now one plus the scalar. And that's going to shift things out and I can go very far in my rotations, in fact I can do everything except for a complete revolution. Tumbling upside down? No problem. I will just divide it by one, no big deal. Okay, so let's explore that. As before, there's mappings, this is goes from quaternions to MRPs. [COUGH] You can inverse this mapping using the quaternion constraint and solve it and end up with these formulas. This is one of your homeworks you'll do with this math and algebra. It's this more ways to apply the quaternion constraint. You need to get comfortable with that. So there's nice elegant quaternion MRP direct mappings, definite singular, every three parameter set by itself has to be singular somewhere, so here is 360s unlike CRPS they blow up to infinity. And if you map it back with a little bit of a trigger to the [INAUDIBLE] you can prove this now becomes tangent phi over 4, CRP was tangent phi over 2. So, CRP, we could visualize small angles as being one, two, three, your row pitch and young angles, essentially, right? This linearizes to angled over 4. So, whatever MRPs you have, you multiply it times four, it's roughly that many radians. Now how good is that approximation? Tangent phi over four. Let me see. I was going to actually run this. There we go, tan function. What I want to do is, if you look at ten function If I go to well, tangent phi over 2. 180 degrees, I'm blowing up, I'm going to infinity, I go off the chart here, right. That's what it act. But you can see that I can be up to almost 50, 60 degrees and there's barely any distinction between the linearized version which is tension x is equal to x and the actual tangent function. So this gives you a quick visual illustration that the CRPs actually lineralized much better then the oiler angels. Oiler angels, 60 degrees, you better have some high order terms in there. Now, if we go here to tangent phi over 4. Look what happens. Really, I mean it's almost linear up to 90 hundred-ish degrees that you can do. If you're going from zero to 90, 100, that's a huge domain that you can deal in a linear if you do a linearized analysis, linear feedback gain analysis which we'll get to. This was actually really valid and if you exceed that domain, if you look at this curve, keys wise, it's almost straight, right? Yes, it's starts to deviate from the blue line but, it's not taking off to infinity, that crazy, all right? So, this is how well these things linearized, they're really really good for large rotations and the linearized equations are quite representative of the non-linear response. They're not exact but you get a very good estimate, which is going to be very, very handy for feedback gain analysis. We can go higher order, actually with a Cayley transform. I've got whole papers published on this, with Hawkins and Jenkins, almost called Higher Order Rodriguez Parameters. It turns out we can take these things to a higher order in the Cayley transform and I can come up with coordinates that are tangent fee over six, tangent fee over eight. And you can arbitrarily bring this closer and closer but you're dealing with more local singularities as you go to higher stoppers. You have six sets of coordinates but six singular conditions we have to keep switching between them so things get a little more, for every gain there's a price. But, you can expand this, so here you see the tangent phi over four part and what happens there. Okay and there's also, if you can do the math, there's ways to relate these CRPs to MRPs, and we can go back and forth. So all these Rodriguez tend to have nice compact direct relationships. We dont have to go to and from DCMs all the time. Just to relate one for the other. DCMs work, but it's nice when you got these really compact, elegant, analytical answers. So modified Rodrigues parameters. We have 1 plus this. That moves it as far out. One question that often comes up is, well if you did 1, why not put 2 there? Right, if we had two, two plus Beta Naught, Beta Naught between plus minus one, that denominator never hit zero. It is real reasons for that all over a sudden, you could do that, then you have a set that never go singular, but if I give you an attitude in that set, those attitude coordinates can all of a sudden represent two different attitudes. So if I give you an MRP of one, two, three, you wouldn't actually know what the attitude is. There's two possible conditions, which is hell. [LAUGH] If you're doing control estimation, then one's a true attitude and one is of wrong attitude. And you don't know which is which. So, that's why one turn out is the best we can do. And you'll see a geometric reason for that shortly as we look at the stereo graphic projection. So Lewis, good question. >> So it's nice that there's no singularity with those but is there a way? >> There is a singularity right now. >> At 360. >> At 360, yes. >> But if you have gone to 190 degrees and you want to find the short way back to zero for controlling, it was easy with the because you just. >> Yes. >> The negative. How do you do it with these? >> I'll get to that. We are getting there so this is just different ways to go from the DCM to here, like with the CRPs, there's some kind of back formulas so if you're coding them. But they all will have issues and now if you're doing a 360 rotation as along as Beta Naught is not minus one then you are fine, so this actually works pretty well. To get one of them, the short rotation, in fact if DCM, as long you pick the Beta Naught that is positive, the positive square root of this you're guaranteed to have the short rotation. So this would actually be a non-singular way to always get to short MRP back, what we will see is there are, CRPs were unique. That CRP and the shadow set on that sphere, both those points projected to the same point in the cube space. MRP's those two points will project to different locations. So there's two possible MRP sets. And they're very equivalent to quaternions. One is short, one is long. This math will actually always give you the short one. If you want 80, there's an ambiguity, but pick one, they're both equally good. All right, if you 180 down or left or right, it doesn't really matter. But the mathematics work out just fine around that.