0:12

The Cayley Transform allows you to input an N dimensional,

Â skew-symmetric matrix and you put it into this math.

Â And outcomes in an orthogonal matrix, or what's really

Â awesome is you put the same math, instead of putting in a skew-symmetric matrix you

Â put in orthogonal matrix and with the same math outcomes skew-symmetric matrix.

Â So the forward mapping, inverse mapping was the same algebra,

Â it just depends on what the input is, which is kind of really cool.

Â So, way back in the day then, actually, I was close to wrapping my PhD.

Â [INAUDIBLE] and I weâ€™re really pondering, can we expand this Cayley Theorem to,

Â the Cayley gave us basically for if this orthogonal is a three by three matrix,

Â proper orthogonal,

Â then the skew-symmetric matrix represents classical Rodriguez primers.

Â And then you can expand this concept of classical Rodriguez primers

Â to multi dimensional attitude descriptions which has applications and

Â structures in different kinds of fields.

Â We were wondering okay CRP's are nice but we know MRP's are so much cooler.

Â Let's face it right?

Â They're so much, lots of really nice properties that come with that.

Â Can we do that in higher dimensional spaces?

Â So this is what we found.

Â We can modify Caley's theorem without the squares here.

Â This is just one.

Â This is Cayley's theorem, and that takes a skew-symmetric matrix and

Â gives you orthogonal.

Â And the same mapping goes back again.

Â Here, the S's, as you will then guess, this becomes our sigmas.

Â This is a formula that will give, if you put in a skew-symmetric matrix of MRPs,

Â and put it into this math, you get to the DCM.

Â If it's a three dimensional space but

Â it turns out this property holds also for N dimensional spaces.

Â So we can use this to define higher dimensional MRP like coordinates that

Â define the orientation or, in this case, the state of a orthogonal matrix.

Â 2:06

You can switch the order, that's nice, but one thing we do lose is,

Â with the classic Cayley, there was the inverse mapping with the same formula.

Â You just put the C in here, and outcomes an S in the front.

Â That no longer works once you go to higher dimensions.

Â And we're doing here too, there's a whole another paper we wrote together on

Â one with going to higher order, to nth order.

Â So we can do what's called higher order Rodriguez parameters instead of tangent

Â fee over four, we can do tangent fee over six, fee over ten.

Â 2:35

But every order introduces more and

Â more singularities or, that you have to account for along the way.

Â So you flatten that curve, you get more and more linear responses, but

Â at the cost of increased number of singular points you have to account for.

Â But there's whole theories and papers on this.

Â So instead of doing the full rotation here, what we also looked at is

Â to do the inverse, basically we can take the square root of our orthogonal matrix.

Â Now what does that mean?

Â 3:02

You get to four by having two times two, right.

Â Attitudes are matrix multiplication, so think of it that way.

Â So we have a matrix W that it squared becomes the DCM.

Â And if we multiply DCM's together so each W is a DCM essentially so

Â this rotation times the same rotation again, gives me the total rotation.

Â So the W the square root operator basically says look,

Â I'm giving you the half rotation DCM matrix.

Â And that's what that means geometrically.

Â Again, once in math, I'm not going to go through details.

Â I'm just trying to give some highlights of what people have looked at here.

Â So if you do this in the square root of diagonal,

Â this is a way that you can do it, matrix square root operation.

Â What it actually manifests itself as is we have this +1 because we know it's

Â a orthogonal matrix and the other one becomes complex conjugate pairs.

Â If it were three by three this is what you'd have as Eigenvalues.

Â There's that 1 + 1 we found in earlier work we did and

Â there's a complex conjugate set.

Â But this W now has angles over 2.

Â And if you do more dimensions, you don't just have one principle rotation angle,

Â if you go to higher dimensions, published a lot on this,

Â you have multiple principle angles.

Â It's like rotation supplemental folds and how does it all manifests.

Â And it really, you have to have some good bottle of Italian wine and anymore and

Â this will make a lot more sense.

Â If you're talking with these hands, kind of like what I'm doing and

Â it'll all look amazing.

Â But so just as we go to higher dimensional spaces, these ideas of principal rotation

Â angles expand but it doesn't just become one as we have a 3D but you get

Â multiple ones and is it not dimensional or even dimension affects all of this.

Â But you can do this, so now I can rewrite and go from the MRPs to this half

Â rotation, forward and backwards, using the classic Cayley formula.

Â And then there's all the symbols.

Â So then we regain all the properties we liked from the classic Cayley, but

Â you get this extra step of doing half rotations.

Â 5:01

So that's been looked at.

Â And similarly, you can get your differential kinematic equations for

Â interdimensional stuff.

Â This is like a MRP rates and how it relates to these Omega's and

Â these whole theories and how to do all this stuff.

Â But if you interested you can look the formulas up and go there and to try and

Â create awareness.

Â And the math you learning and the projections you learning and

Â this principle rotation stuff.

Â We typically apply to three dimensional space but

Â this whole mathematical theories of taking these ideas.

Â To n-dimensional space as well, and

Â Cayley's theorem is kind of at the heart of that.

Â Jordan. >> I don't know if I missed this, but

Â if you go back one slide.

Â >> Yeah.

Â >> In this classical Cayley transform, is that S,

Â that's not the skew-symmetric MRPs, right?

Â >> This is the skew symmetric one But it's the formula,

Â it's like the Cayley, the classic.

Â And you can reverse the orders and

Â forward the mapping between W and S, it is just like regular Cayley.

Â >> Well, but W is the square root of DCM, right?

Â >> It's the half rotation, yes.

Â >> S the square root of- >> No, S is the full rotation.

Â >> Okay. >> And then this gives you this.

Â 6:03

And then W squared gives you the actual rotation.

Â So, that's how the mathematics works out and you can go in Math Lab and

Â place some numbers, plug it in and you could prove this to yourself once you

Â see the pattern but Iâ€™ve kind of where those things go, good.

Â 7:00

So an open research question, if any of you has too much spare time on

Â the weekends or the evenings is how do we actually switch from MRP sets shorts to

Â thew long for we can avoid singularities, how do we do this in higher dimensions?

Â Nobody has quite figured that part out yet.

Â The geometry, the mathematics of it.

Â Thereâ€™s always new nuggets, if you kind of trying to thinking yourself to do or

Â curious, or if you work in this area.

Â And he was using some numerical tricks to do it, but

Â he was wondering there should be a nice analytical angle.

Â I agree. I just

Â haven't had time to delve more into this.

Â Definitely exists.

Â So all of this is ongoing work.

Â I'm just trying to show you some elements.

Â These things kind of come in chunks as somebody gets excited about this and

Â makes more stuff.

Â So unsolved problem.

Â