Now, we're going to go a little bit more quickly through this. You've seen the Cayley Transform. The Cayley Transform allows you to input an N dimensional, skew-symmetric matrix and you put it into this math. And outcomes in an orthogonal matrix, or what's really awesome is you put the same math, instead of putting in a skew-symmetric matrix you put in orthogonal matrix and with the same math outcomes skew-symmetric matrix. So the forward mapping, inverse mapping was the same algebra, it just depends on what the input is, which is kind of really cool. So, way back in the day then, actually, I was close to wrapping my PhD. [INAUDIBLE] and I we’re really pondering, can we expand this Cayley Theorem to, the Cayley gave us basically for if this orthogonal is a three by three matrix, proper orthogonal, then the skew-symmetric matrix represents classical Rodriguez primers. And then you can expand this concept of classical Rodriguez primers to multi dimensional attitude descriptions which has applications and structures in different kinds of fields. We were wondering okay CRP's are nice but we know MRP's are so much cooler. Let's face it right? They're so much, lots of really nice properties that come with that. Can we do that in higher dimensional spaces? So this is what we found. We can modify Caley's theorem without the squares here. This is just one. This is Cayley's theorem, and that takes a skew-symmetric matrix and gives you orthogonal. And the same mapping goes back again. Here, the S's, as you will then guess, this becomes our sigmas. This is a formula that will give, if you put in a skew-symmetric matrix of MRPs, and put it into this math, you get to the DCM. If it's a three dimensional space but it turns out this property holds also for N dimensional spaces. So we can use this to define higher dimensional MRP like coordinates that define the orientation or, in this case, the state of a orthogonal matrix. And dissolve in the system if I connect, this. But we have to go to squares. You can switch the order, that's nice, but one thing we do lose is, with the classic Cayley, there was the inverse mapping with the same formula. You just put the C in here, and outcomes an S in the front. That no longer works once you go to higher dimensions. And we're doing here too, there's a whole another paper we wrote together on one with going to higher order, to nth order. So we can do what's called higher order Rodriguez parameters instead of tangent fee over four, we can do tangent fee over six, fee over ten. But every order introduces more and more singularities or, that you have to account for along the way. So you flatten that curve, you get more and more linear responses, but at the cost of increased number of singular points you have to account for. But there's whole theories and papers on this. So instead of doing the full rotation here, what we also looked at is to do the inverse, basically we can take the square root of our orthogonal matrix. Now what does that mean? You get to four by having two times two, right. Attitudes are matrix multiplication, so think of it that way. So we have a matrix W that it squared becomes the DCM. And if we multiply DCM's together so each W is a DCM essentially so this rotation times the same rotation again, gives me the total rotation. So the W the square root operator basically says look, I'm giving you the half rotation DCM matrix. And that's what that means geometrically. Again, once in math, I'm not going to go through details. I'm just trying to give some highlights of what people have looked at here. So if you do this in the square root of diagonal, this is a way that you can do it, matrix square root operation. What it actually manifests itself as is we have this +1 because we know it's a orthogonal matrix and the other one becomes complex conjugate pairs. If it were three by three this is what you'd have as Eigenvalues. There's that 1 + 1 we found in earlier work we did and there's a complex conjugate set. But this W now has angles over 2. And if you do more dimensions, you don't just have one principle rotation angle, if you go to higher dimensions, published a lot on this, you have multiple principle angles. It's like rotation supplemental folds and how does it all manifests. And it really, you have to have some good bottle of Italian wine and anymore and this will make a lot more sense. If you're talking with these hands, kind of like what I'm doing and it'll all look amazing. But so just as we go to higher dimensional spaces, these ideas of principal rotation angles expand but it doesn't just become one as we have a 3D but you get multiple ones and is it not dimensional or even dimension affects all of this. But you can do this, so now I can rewrite and go from the MRPs to this half rotation, forward and backwards, using the classic Cayley formula. And then there's all the symbols. So then we regain all the properties we liked from the classic Cayley, but you get this extra step of doing half rotations. So that's been looked at. And similarly, you can get your differential kinematic equations for interdimensional stuff. This is like a MRP rates and how it relates to these Omega's and these whole theories and how to do all this stuff. But if you interested you can look the formulas up and go there and to try and create awareness. And the math you learning and the projections you learning and this principle rotation stuff. We typically apply to three dimensional space but this whole mathematical theories of taking these ideas. To n-dimensional space as well, and Cayley's theorem is kind of at the heart of that. Jordan. >> I don't know if I missed this, but if you go back one slide. >> Yeah. >> In this classical Cayley transform, is that S, that's not the skew-symmetric MRPs, right? >> This is the skew symmetric one But it's the formula, it's like the Cayley, the classic. And you can reverse the orders and forward the mapping between W and S, it is just like regular Cayley. >> Well, but W is the square root of DCM, right? >> It's the half rotation, yes. >> S the square root of- >> No, S is the full rotation. >> Okay. >> And then this gives you this. And then W squared gives you the actual rotation. So, that's how the mathematics works out and you can go in Math Lab and place some numbers, plug it in and you could prove this to yourself once you see the pattern but I’ve kind of where those things go, good. So there's lots of stuff actually, because an open question was sent to me, well about 10 years ago now. Eigenvectors stuff from trying to program this off and we've been using your formulas and doing this. But they have structural components that are revolving. So they are running into issues when they do this. Why is that? Well we know MRPs blow up if you don't switch them to shadow sense. So an open research question, if any of you has too much spare time on the weekends or the evenings is how do we actually switch from MRP sets shorts to thew long for we can avoid singularities, how do we do this in higher dimensions? Nobody has quite figured that part out yet. The geometry, the mathematics of it. There’s always new nuggets, if you kind of trying to thinking yourself to do or curious, or if you work in this area. And he was using some numerical tricks to do it, but he was wondering there should be a nice analytical angle. I agree. I just haven't had time to delve more into this. Definitely exists. So all of this is ongoing work. I'm just trying to show you some elements. These things kind of come in chunks as somebody gets excited about this and makes more stuff. So unsolved problem.
