Addition, subtraction hopefully boring materials. I see some yawns, thank you right on cue, okay. because now it goes [SOUND] we've done this many times, right? There's always a good old method of mapping into DCMs, we multiply out the DCMs, make sure you transposing the right the DCM and you can addition subtraction anything you wish. And then at the end you get a DCM and we know formulas to pull out, whatever coordinates we wish, so we can even mix coordinates and add them. This always works but if you only deal in MRP space, you don't want to deal with the computational overhead of having a three by three, a mapping to three by threes. Then three by three math, multiplied out and then extracting and mapping back again. This is computationally faster and there is a direct MRP addition property that we have here. Like with the CRPs. But with the CRPs, if I added a 90 degree rotation and a 90 degree rotation about the same axis, the answer is 180 degree rotation, my answer will blow up. And there is nothing that you can do about that, right? Because CRPs are unique. MRP's though, I have options. They blew up at 360, so if one, if the prime, if here we have a 180 degree rotation, about B1, and I add a 180 degree rotation, again about B1. The answer has to be a 360 degree rotation about D1. And this MRP set should blow up, in fact it does because your denominator goes to zero. But can somebody see a way to avoid this now? Jordan. >> Do one rotation by 180 and the other by negative 180? >> Yeah, because we're adding 180 and 180, but that 180 we're adding is just one of the possible ways to describe that rotation. Equally accurate is also minus 180 and 180 minus180 gives you zero. And that's in fact, the other, the short rotation, right? So if you did this in code, I wouldn't just write this. But you can do it in a non-singular way with a single if statement. And that says, check if the denominator is near zero. And you don't even have to go down to tens to the minus14 or something, you know this works perfectly well. If it does anywhere close to being 0 you know you're close to the 360 mark. And the only thing you have to do then if this denominator is near 0, you switch either. Well you want to make sure you switch one of them to the other set. I wouldn't just say the first or the second, there is some logic you want to build in because let's say somebody, some joker gave you 0+360, well they have to give infinity actually. That would still be difficult. [SOUND] But a really computer scientist found out a way to get this zero, you might want to check the norms in there as well. But practically, to get a singular answer you would have and there not infinity to begin with, you would have two distinct sets you should be able to really flip either to the alternate set. And then the sum will give you the short rotation. So if this goes to near zero, you're getting something that is going to give you the wrong rotation, so that's the one switch. And then with this addition property, I get it! I am done! Am also showing subtraction which is an explicit formula, but here is a cool thing, we do solve that. If I need to Sigma BN, and I have sigma NB, I go from BN to NB by just doing the minus sign. So if I have to take this 180 degree rotation, and then subtract a 30 degree rotation to get that relative rotation we're looking for, subtracting 30 is the same as adding minus 30, [COUGH] right? So with the same addition formula, you can actually do addition and subtraction. It's just, if you want to subtract that you don't feed in Sigma, you feed in minus Sigma. And that would be hey, 180- 30 gives me the relative attitude of 150, that you would have. So, there's lots of little details on how you can do this. And this makes it actually quite powerful, with a free parameter set, it's very fast, if you're on a fly computer, limited CPU capability. This is a way that you can implement it and there's little if statements that do this in a completely non singular way and you can do that. If you don't trust your math always double check here. This will definitely work. DCMs are non singular but you got a lot of matrix math to do and arithmetic errors that can accumulate. This gives you a lot less stuff to do, so it's very fast. Any questions on addition and subtraction? Okay, good. I say at this point I hope there's not there's not many questions because its kind of the same stuff we've seen over and over again. All right, it's just a little It's just a little different formula, different nuances and attitude coordinate sets.