Now lets talk about, a very powerful property of these MRPs. And that, Their shadow set. The shadow set is the properties of these Rodrigues parameters, if you don't use the original quaternion set, but you use minus betas, right? So instead at the point here in the surface, you go to the entry point, and you use the other description. With the CRPs we reviewed, they mapped to the same point. And the short, long, all give you the same answer, there's nothing you can do otherwise, that's it, that's the behaviors. But with MRPs, if we just had this, it's minus beta over minus beta naught, the vectorial of a scalar, the minus is cancelled, it gives you the same mapping, that's what we had with them. If you've got one in there, you get a different number. And there's very particular properties. So like with the quaternions with there's two possible sets and they're non-singular MRPs individually are singular, they blow up at 360. But there's also two distinct numerical sets. And one set of values will define the short rotation, and the other will define the long, that's what we're going to call the shadow set. So, instead of long and short, it's kind of the shadow thing, that's it. That was made awhile ago. And so, if you do this with the betas, all we had to do is reverse the sign of the betas. Whereas with the sigmas, and you can prove this to yourself quickly with a little bit of algebra, what you end up getting is the mapping of here to the others is nothing but minus beta i over the norm squared. So, let's look at this. The typical surface where we choose to switch is sigma squared = 1. If you think of the definition of sigma i as being the ei times tangent phi / 4, the tangent 1, where does it go to 1? And that happens at a 180 degrees. Then you got tangent of 45 is 1. So at 180 degrees, the norm of the MRP set is simply 1. So in the three-dimensional MRP space, that one surface, which is again, it's a unit sphere, is very important. It's not a constrained sphere like with the quaternions where you have to reside on it. In fact, I'll show you a picture we reside typically inside. The zero orientation is simply 0,0,0 MRPs, very nice, right? But you can move inside of it. So, what we can look at is right down here, we can see if the norm of the MRPs is less than 1, I am describing a short rotation. So now, Tebo was asking earlier about plotting an angles with the MRPs typically, so I don't want him to blow up. I can't let him go up to 720, and higher, and all this kind of stuff. Typically, I always plot them within the plus minus 1 line. And so I click the norm, and you see the MRP description. And at some point if they describe a long rotation, I would tend to flip them back to the short rotation, so I've always got that. So I show them as a discontinuous set but I'm always describing the short rotation. If it's equal to 1, if the norm is equal to 1 of the MRPs, then I am 180 degrees off. And this one, actually be handy in feedback control as well. Because your attitude errors cannot grow infinitely large, right? The worst attitude error you can have is being backwards compared to where you're supposed to be pointing, that's it. Any more of a rotation and you actually reduce your error. So in MRP space, my attitude measure norm is bounded by 1, which is really cool from control point of view because we can give a very clean clip on that's the worse that can happen. And you will see that happening later. If your MRP norm is bigger than 1, nothing bad happens. If it's 1.01, okay, you're describing something that's slightly longer than 180 but it behaves still perfectly fine. It's not until you get close to 360 that it starts to go off to infinity, okay? And this has an impact on how we immigrate this things but this is the shadow set. This is kind of a visualization of it, so think of it this way, I'm starting out originally at the origin, so following the grey line. And at the origin, b and n are identical, I've got 0, 0, 0, that's it. Now you're doing some tumble, wobbling around and doing stuff. At some point here, I almost leave the unit sphere, but not quite. So I was almost backwards, but I've recovered, we came back around. At some point now, I've tumbled past 180, that's this point. That's where I punch through that MRP units sphere. Again, it's not a constraint, it's just a convenient boundary. And I know from here onward, my description as a completed revolution to 360, it's describing the long rotation, and eventually it will go singular. But because we have this analytic mapping, with quaternions it was minus beta to go from one set to another. With MRPs it's minus sigma, but scaled by the norm. If you switch precisely on the 180 degree surface, it's simply minus sigma. It's exactly like the quaternions but having said that that's nice for analysis in your code you never hit that MRP equal to 1 surface precisely. So just implement this full, get the norm might be 1.000001, fine, divide by that now you lose absolutely no precision as you switch from one set of description to another set of descriptions, right? So it's nice that we have a general 1, it's nice that we switch when it is equal to 1, that's good for analysis. But in your code, you don't want to iterate in your time steps, I went too far, I went too little, when did I hit that surface? You don't need that, just let it punch through the surface, nothing bad happens, but then we switch. So this is what happens here because we've got this convenient mapping. If you have 1 trajectory without integrating again, you just simply use minus sigma i over the norm squared, and you can immediately plot the other description of the same rotation. And that's what the black line does. So as the original description exceeded 180 degrees, and then went off to close a revolution, the other description that was the long way all of a sudden becomes the short way. So, MRPs and shadow MRPs, it's kind of which one's long and short it's just a matter of perspective. I tend to call the shadow set the long set. But really in the code you don't distinguish between a shadow set of parameters, and then short set you just have sigmas. And if the sigma norm is bigger than 1, I'm simply flipping it back to the other set. So, I switch between two of the descriptions. And then as I complete the revolution, you can see this line comes in, and then goes to the origin, perfectly well-behaved, I complete a 360 revolution, and I never hit a singularity. So I can't do that with one set of MRPs. Any three parameter set is singular somewhere. But I can do that with both sets of MRPs because their singular behavior is polar opposites. The 360 point is bad for one set but brilliantly good for the other set. And you just pick with the short rotation. That's how MRPs become, the combined MRP and shadow set. Both descriptions, become a completely non-singular way, to describe any orientation. But we have to deal with this discontinuity in your description, where you switch from one set to the other set. And it's roughly minus sigma but it ends up being minus sigma over norm squared.