Let's talk about modified Rodrigues parameters. So Spencer, do you remember the formula there? We had a Sigma I and how do we map from the oiler parameters to these? Russell? Sorry, I'm off on my names today. Russel. How do we map from Betas to Sigmas? >> I over 1+ Beta Naught? >> Yeah, that's it. Because Beta Naughts have to have values between what number and what number? >> Zero and one? >> Not quite. >> Negative one and one? >> Negative one and one, right. Zero to one, Kyar, zero to one means what for Beta Naught? >> One in the sense- >> If you see the Beta Naught have a value not between zero and one, what does it mean about your description? >> [INAUDIBLE] >> Yes, you have the short rotation, right. So that’s was Nick probably thinking and you can always switch your quaternions such as you do describe the short rotation. When you're Beta Naught is equal to 0, you got the 180 in that case, 0 minus 0 is the same thing. Both sets describe the 180 rotation. So, specifying that would work but now you only have one of the sets and you have to switch between them. So generally Beta Naught is plus minus one. Plus one is good I'll divide by two and I think crazy minus one makes me divide by zero and there is where this will go to infinity. So this is a good review now these MRPs. If I just tell you. I don't remember your name. >> Ben. >> Ben. If I tell you the body frame and the inertial frame happen to be identical right now. Is the MRP description singular or non-singular? >> Non-singular. >> Why? I picked it to be non-singular. >> Okay, is there another option you could have picked? Picking implies there's a choice. >> Singular. >> Right. It could have actually been both, in that case. And that's the weird part all of a sudden, that it beginning, God this gives me a headache. What's going on here, you know? So this is now, we're getting into attitude parameterizations. When all of the sudden it doesn't just matter, how I pitch 90 degrees or not. It actually matters, well yes, you and that orientation, but how did you get to that orientation? It matters as well with these descriptions. And if there's nothing else in the system specifying it, you get to pick. And I would also pick non-singular, in that case, and say it's identical, it's a zero rotation. If you claim this is a 360 rotation, it is singular. Who would choose singularities, right? If you tumble again, we're at 720, now what happens to our stuff, Andre? >> It's a still singular. >> At 720? >> Yeah. Actually, no because it'll be at plus 1 again, so. >> Yeah it's a plus 1 again right? And remember so quaternions, I can measure, have some measure of how far have I tumbled but only up to one additional revolution. Beyond that, it repeats and it goes back to plus one again and then you can't distinguish, did you do 360 or 720? Those are mathematically indistinguishable just by looking at your coordinates, right? So good, those are all the little details we want to get together. And the same thing happens now here. With these Betas. So we've picked one set, good. Now there was also a, yes Jordan. >> Are there sets that keep track of all of your rotations? I assume that there would be, but. You got a cable or something that makes sense [INAUDIBLE] >> You could use, that's a great question. Nobody's ever asked me that one, so that's good extra star for Jordan. Okay, let me think live, I would take a look at principal rotation parameters. You've got differential kinematic equations. There are ambiguities as you hit zero and other stuff but it's most about the axes. You could numerically look at this history and if I'm rotating about one once I've gone through the zero orientation coming out again. There's ways to track if it's a continuous tumble. Then you can just integrate that fee and keep it growing infinitely large and then you have a sense of how far you've tumbled. >> Okay. >> So I'd have to look at some coordinates like that. With MRPs quaternions, we don't get that. Just how they're defined. We get up to one tumble. But no, good. So this is the mathematics of it, right? If we wanted to relate them, the sigma Is were ei tangent phi over 4 which linearizes roughly to ei. Phi over 4. So that MRPs you can think of just, roughly, angles over 4. You all pitch roll over 4. For small rotations that works out pretty well. They actually linearize extremely well even for large rotations, which is a huge benefit. Let me do feedback gains, and you try to find which stiffness which stamping to throw in to make it critically there, we often linearize the close of dynamics and then we pick this stuff. Well, if you linearize them into MRPs, it's an answer that works really nicely, so that's a good thing. But you can see at 360 over 4, you get 90. So tangent 90, that's where it blows up. They also go off to infinity. Robert, what was the geometric interpretation of these. There was also stereographic projection, we could do. Do you remember anything there? Where was the projection point? Where is the projection plane? Let's see, back row. Is that, so that's Russel all right, David? >> Right. >> Okay. >> I'm not sure either. >> Go down the row. >> Brett. >> Yep. >> Was the projection plane at the origin, and the origin was still at zero? >> Almost. Let me get a different color just so this shows. So your projection plane is here, let's pretend that's a plane, not a bunch of scribbles. So, Beta Naught equal to zero. So,we move the plane from Beta Naught is equal to one, one over to the left, right. That's there. If the projection point is here though, on that plane, you wouldn't be able to, everything would intersect in the origin, and all your coordinates would be zero. That would immediately cause issues. So where was the projection point? Anybody remember? Yeah, go ahead. >> Was it at the left point where the hyper sphere intersects [INAUDIBLE] >> Yep. Everything moved. The plane, this is one trick to remember, the plane the way we defined it, was always plus one away from the projection point. You could have made it plus two, plus three, it gives you different coordinates but they're just scaled versions. Then this new coordinates are three times MRPs, yeah otherwise exactly the same behavior. So there's nothing really special about doing that besides having a factor 3 in there. So the projection plane plus 1 away from the projection point, that's just going to normalizes that. So you do this, now we use this 90 degree rotation point we had earlier, you connect the two points, right? Now you get your interface. This is the part that's sigma I, these coordinates here this was Beta Naught, Beta I, and again, like in the homework, you can do similar triangles to prove these projections. But that's the way to geometrically interpret it, right. If we go all the way over, if am up here, what is my body doing? Is it Nathaniel or Nathan? Where am I, the points up there? >> 180 degrees. >> Good, okay, you've start to interpret this thing. 180 degrees and the upside down tumble 360 and this angle is over two, that will be here. So if anywhere close to here, these things become essentially co-linear with the plane, and you never intercept. That's where you blown up to infinity, that's kind of a geometric interpretation of that. Good. That was a quick review, over some of the topics we've covered.