If we have to add and subtract them, we know with these formulas we can go ahead and to map them all to the DCMs. If this is the rotation about which you rotate four degrees to get to B. And then there's a different rotation 45 degrees about a different axis that gets you from B to F. You map it to DCMs, you carry it out, right? And then from the DCMs, you can extract whatever you wish. You can go back to principal rotation vector stuff, or you can do Euler angles, whatever you wish. So we can always do that. As we've seen with the symmetric Euler angles, there's also a direct formula that gets you from one set to another and gives you the sum. And I'm not deriving that here but those are given here where you look at these, these two summations are there. So that's kind of nice. If I have the first set of rotations with the vectors and the angles, then I can sum them up without having to go to DCMs and get the answer. However, does somebody see an issue with this math? Would you put this as is on a flight computer? Charles, what do you think? >> Well, the answer [INAUDIBLE]. >> No? What's the one term that should scare you to death if you're a coder? Matt. >> That sin theta over 2. >> Right, I'm dividing by something, right? >> There a trig identity there then? >> There's a sin phi over 2, what do you mean a trig identity? >> I'm talking about in the upper line. >> Here? >> Yeah. >> What do you mean, is there a trig identity? >> Yes. >> This is a scalar, this is this one, and then you have a dot product between two general vectors. There is no way you would have a trig identity that would rewrite this with this part. Without this part, yeah, there are probably trig identities and how to use sums and subtractions of angles to do this. But with this, this is as compact as you will get it. So here, this is the big issue. So we talked about earlier singularities. If we have zero angle that's a problem, because all of a sudden those other things are ambiguous. So the individual rotations may be non-zero. I'm adding a 45 degree rotation about N3. And then the second one is a minus 45 about n3. Both of them are well defined but the sum is a zero rotation. And that's what will happen here. Or you can have a 90 degree rotation plus a 90 degree rotation. Which means actually both of them individually are perfectly fine. But the sum is 180 degrees, which also the sine has issues with. So you can see right away that's where the singularities are manifesting. So if you use this formula, be very careful you don't hit the zero 180 kind of stuff. Otherwise, you have all these ambiguities. I would probably tend to go more this route here, just for dealing with singularities. Then I get this and if it's a zero rotation, then I get an identity matrix or I would pick variables that just aren't singular. But, so, addition, subtractions, like before, matrix method and we're not deriving this, but it's in the book. You've got this also in a compact direct addition formula, which might be handy, it's analytical. But, it is also singular by itself. Subtractions, ditto, same thing. Now you guys are experts. Once we can add, we fundamentally know how to subtract. We just have to do transposes. And then just develop [INAUDIBLE] it gives you a similar looking equation but, we still have these glaring singularities there. We know these are a singular set of four parameters. The principal rotation vector, right now we've always talked about four parameters. The principal rotation vector is simply E hat times big V. And in the homework you will see this too, that's what I'm calling as the gamma vector basically. So I'm taking a four dimensional set and I'm reducing it to a three dimensional set just by taking the E hat. Now I'm making it a non-unit vector. But it's just one set. I'm back to three parameters. Any three parameters have singularities. Which will be 0, 180, that's what we have in this case. And in fact we're not deriving them, but if you needed them these are the differential kinematic equations. We talked earlier, they will relate your omega between b and n to the attitude description between b and n. And this gamma vector set here has this, the b matrix we talked about earlier is this set. And sure enough you see stuff like 1 over phi squared. So we know zero is a trouble maker. It has to be because there's infinity of answers. And also actually upside down 180 causes issues here with the cotangent function that we have to account for. So that's where the things mathematically manifest themselves. If you need the inverse of this matrix people spend quite a bit of time doing all this algebra, and you can come up with this formula. So it's nice that we have nice compact forms of this B inverse. But it's not a DCM, it's not just your orthogonal part doing a transpose operator. It takes quite a bit of extra math to solve this analytically. So we have these. I don't use this for my tracking error description. It has a tracking error, drive B towards the reference, I want my attitude errors to go to zero. Why would you pick an attitude description that goes hog wild at the stage you want to reach, right? So this is a great building block, computer vision uses these a lot. They love these things, you see them all over the place. And also it's a good thing to know about. But from the aerospace application and control, and particularly we don't tend to use it in this form. So what we'll start next is looking at other parameterizations that all build on these properties. Which are four coordinates is a four dimensional parameter set. There are four possible answers, so they are not unique. They are singular, all these issues. But we can now exploit these behaviors and construct very cool, elegant, different attitude coordinates and different kinds of behaviors. That's where we we'll go next. Okay. So thanks. If you haven't turned in your homework please drop it off up front. Yup before I wrap that up. Great and I'll see you guys on Tuesday. Yes? >> I have something.