Let’s do a quick overview, highlights, cliff notes, whatever you want to call it of Euler Parameters or Quaternions. Let me learn some names, so grey shirt. What’s your name again? >> Nick. >> Nick, what’s the difference between Euler parameters and Quaternions? >> They're the same. >> Yeah, basically array of different ways, different theories. But basically, they give you the same thing. There is a field called quaternion math, it's like complex numbers. Instead of just real and one complex, you have real and three complexes. There, the quaternions can have non-unit length. So sometimes you see people talk on their literature about unit quaternions, they're trying to be extra specific for the audience. If we talk to an attitude persons, talk about quaternions, 99.999% of the time they know exactly mean unit quaternion. All right so again, different fields approach this stuff. You got atomic physics, there are other ones, they all deal with this kind of orientation as well as different notations. So good, quaternions. Let's see. Andrei- >> Yup. >> How many quaternions do we have? >> One, four. >> One or four? How many coordinates do we have? >> We have four coordinates. >> Right, and in this class we're going to use the Beta notation, right? So we have Beta zero, Beta one, if I can write, Beta two and Beta three. So, Spencer. Of these, with the one, two, threes and the zeros, which one are always the same number? And with different notations, which ones might be different? >> [INAUDIBLE] >> Different notations, yeah. Some people do queueus, for example. >> [INAUDIBLE] >> Exactly, so always look for that. If you're dealing with quaternions, it's a fact of life. There's huge different literature, different fields. The parts that are always the same are the vectorial part and that's what we're calling here. In fact, on the last slide, I'm going to call that the epsilon vector. So sometimes, in the control, the beta 1, 2, 3 or the q 1, 2, 3, that's the vectorial part. Now, Robert, why would we call that the vectorial part of the quaternion? Where do you think that came from? If you think of the basic definition of quaternions. >> Use the rotation axis [INAUDIBLE] versus to calculate [INAUDIBLE] >> Exactly. E one, two, three which comes from the E hat vector, all right. And this could be in B or N frame if it's the B relative to N attitude. This has, this relates, I mean this is influenced by these vector components, so it matters about which axis have rotated ten degrees, alright? And the rest of it was sign phi over two and that's the same. This one is called the scaler part of the quaternion unit and it only cares about how far you have rotated. That was a benefit of the principal rotation parameter, also a four parameter set. But one of them was explicitly a scalar that only cared about how far, and that's very handy for control applications. I can specify, I got within one degree. I often don't care about which access. I just need to know how tight was my cone. With quaternions you do the same thing, but you just have to look at the beta not part. That's your scalar part all right? So there's that slight distinction that we have here, excellent. Back row, if you don't mind just closing the door as wel, l just for noise and stuff I appreciate it. Good okay, so let's the classic questions. Are these things unique? Sorry, what was your name again? >> Kyle. >> Kyle. >> No. >> Okay, If not unirque, how many sets are there? >> Two? >> And then Kevin, right? How are those fifth from MRPs mathematically related to eavch other [INAUDIBLE] >> So right, negative sign. It's a really easy mathematical mapping if you have one set and you want to consider the other set, we just flip the sign, that's it. So Mariel, what's the difference between, you know geometrically speaking what's the difference between one EP parameter description and the other. [INAUDIBLE] Yeah. What does it geometrically mean, having- >> [INAUDIBLE] >> Not quite. You're on the right track. Let's look at the zero, Bill, what is the zero rotation in all the parameters? Brian not Bill. >> [LAUGH] >> Close, I got the Bs right. >> Zero rotation? >> do you remember? >> As can't written down upside down I'm not sure what you'd say. >> You've got the formula right there. Cosine of 0 is? >> 1. >> There you go, right? 1000, sines of 0 will all be 0, 1000. That's one possible answer. Nathan, what was the other quaternion set? >> Okay, what's the different between 1000 and minus 1000 ck? [INAUDIBLE] >> Okay, that's the interpretation of the constraints surface that you're talking about. Good we're about to get into that. But there's a fundamental difference those two descriptions of this orientation. Mandal? >> [INAUDIBLE] >> Right, there's a short and a long, that's it. So this description now gives us the option of describing a short way around or the long way. With the DCM we didn't have that. The DCM was unique. For an attitude, that's only one DCM. That's it. Which is actually very nice. But that's it. Principal rotation parameters, there were four possible sets and there was always a short rotation or you could go the long way around, or you could spin about the opposite access, right. Now we don't have to access the options here in all the Euler parameters. But we do have always a short description and the long description. This impacts you when you use them for feedback control because you don't want to feedback an error of 359 degrees, you would rather do minus one, right. Just nodge to your left don't spin the way around. And it impacts in other ways but both sets completely non-singular. You will find differential kinematic equations we derived. Our good for either set, it does no mathematical distinction between what's one and what's the other, right? because what happens as I do to Andrew, let say we do a multi-revolution thing, you start out here, this is 1000, you do a revolution, now, you have a minus 1000. And if I do another revolution, I've done two revolutions now, what's my altitude again? >> [INAUDIBLE] >> Now you're back to one, right? So that's why you could keep track with Euler parameters if you've done one revolution but I cannot mathematically distinguish if I've done five or six or seven revolutions right and then the descriptions repeat. You just oscillate the two between, two possible options so you could only keep track of one revolution if you want to. So that's a big part of this as well. Okay, so they are definitely not unique. We have two possible sets, they're non singular. They always work and easy mapping between one set and another is this attitude is equivalent to this attitude. It's just a sign flip. How simple is that? That's cool. Now so in a control, let's say, you integrate these and at some point, you want to switch. Let's say, it's tumbled past 180 degrees. I don't want to describe an error of 200 degrees. I would rather do the short rotation. We could use this math and get there. How, sorry, last row, what was your name? >> Brett. >> Brett, how, if you look at these, Euler parameters, how do you distinguish if you're doing the short rotation or the long rotation? There's one of them that kind of gives them away. >> Beta zero? >> Beta zero, yes. The scaler one Coz if you 280 degrees 180 over two is 90 cosine of 90 is zero, right? So we start out of one, one Beta Naught of one, means I have no rotation then I'm upside down Beta Naught goes to zero and if I complete a revolution it goes to minus one eventually, right? So good we got all of that going, that's the one there. I mean it's mentioned upside down. Now CK you were talking about this hyper surface. Tell me more about that, that sounds interesting. >> Okay so it's, it has to be we have a constraint on the ought to have >> Be more precise. Right we have to square. >> The square against. >> Yes, this morning had a question about this, so let me just be precise here quickly. We often say adding up to one. It's implied we mean the norm squared, do you have to square each and add them up right? But that's the mathematical script, no wait that's a two not a three. Beta two squared, Beta three squared equals 21. This is your constraint. All right. THis is a four parameter system. It's always a three degree of freedom problem. So we have one constraint. But this means now also, when you integrate these differential equations I showed you last time, when we had the Beta dot equal to this B matrix times Omega. If you integrate this, every time step we tend to reset them, cause they'll be off by just 10 to the minus 13 or something really small depending on your integration steps, but you have to re-normalize them again. With DCMs there’s also a process of how to reorthogonalize a matrix, it’s just a little more involved. With Euler Parameters, pretty straight forward. Take the, just divide by the norm and you get it back to a normal length to within numerical precision. Good, so we had to apply that there but also deriving and as you said, this really describes a surface. A hyper surface where if you're on one point there is another point on the opposite side that describe the same orientation. So, as you move the trajectory around, you really, your MRP has to evolve in this unit surface and the anti point is the opposite set which can be a shorter [INAUDIBLE] depending on how you rotate it that kind of becomes interchangable. Good, I'll go there first. Okay, Elmer Spencer. >> [INAUDIBLE] >> Yes. >> Is there any way [INAUDIBLE] >> This part. You look at the coordinates. So it's kind of you think about it as XYZ space. In that case it would be the x coordinates. If the X coordinates is positive I am on the side has the short rotation. Everything else, the other have the hemisphere has the long rotation. It’s just hard to think of that in 4D space. But it’s mathematically equivalent. Louise, yeah? Okay. >> If you're integrating those and trying to [INAUDIBLE] strength and if you're doing something like RK 4 you would you enforce that at the end of every time step or at the end of every [INAUDIBLE] >> Good question because we're going to get into this. That's a subtlety but that's a really good question. Let me take that off for now. So the question is essentially, you've got your current set of Betas. All right and then you're using an R K four, how am I going to do that symbolically? It computes a K one sets that's a function of the current stuff, and then you have a K two, a K three, if I can write, a K four, all right. You blend them together and then the final answer ends up being these things summed. Yes, angle K4 and there's a scaling parameter that goes here, right. And this is how you blend them together. Typically, what we do is at the end of the integration that's when I would normalize them again. And you might have really small math errors that build up in between. If you wanted to, you can try and doing there but you'd have to have very large time steps. That is what actually matter. So the renormalizing and just make sure this is always a valid unit for quaterion constraint. Nothing too crazy would happen here. But this is a good question, because as we go to other parameters, especially the modified Rodrigues parameters, this thing will matter. But yeah, practically, I'd go to the next time step, integrate there, then I'd renormalize, and that seems to work pretty well. If you did research and you're really picky, am I doing it right? There's different methods for integrations that you can check. You can check energies, you can check all kinds of stuff to make sure what's my numerical error and what time steps and do these different local discrete approximations make sense. Good, okay. So that was a quick overview of quaternions, they're built on principle rotation parameters. And we went through all those sets including the differential kinematic equation and highlight here, this is something you drive in the homework spot in the end, if I buffer the usual three by one with a zero upfront making it four by one, the only reason we do that is this matrix happens to be orthogonal, right. That was nice, then you can easily invert it. You can re-write this needed. Wait I need to go. There you go, you can go into this form. There's different ways to write this. Typically it's the same. This is now a 4 by 3 matrix. There are some properties here, that are useful. You can easily, this will take two minutes just to double check yourself. But we'll use them later on. Just wanted to highlight them here. And again, sometimes you see the quaternion math written out as what's the differential equation for the scalar part of quaternion and what's the differential equation for the vector part of a quaternion. For example, let's just ahead. The control, let's make sure this actually makes sense. In the controls we typically actually only use the vector part of the quaternion. So if I drive epsilon to zero, I'm trying to get the spacecraft to realign with the inertial frame, a simple regulation problem. And I drive epsilon to zero, am I guaranteed that I'm pointing in the right direction? Shaila, what do you think? I see a quizzical look and I though I would hear your thoughts. >> I don't know, I would say no. >> Well no, it's 50/50 right. But let’s talk it through, what happens if this part goes to zero? What must a forth or the paramater be in that case? >> Four. >> Right. And it's that the zero orientation. >> [INAUDIBLE] >> Yeah. Negative one is it also a zero orientation. It's the same orientation but is it the same path to this orientation? No. And that's where we're going to get in to also now with Rodrigues parameters starting today, wrapping up on Tuesday. Often some, the path to this orientation matter. So if you just go out, so one of them if you just in a control drive this to zero, without looking at what quaternions set you're using, you might rotate all the way around. And you get there, perfect stability, it is not a very nice performing control, there's a much faster path to get there, right. So even in quaternions sometimes people switch. They look at the description and the attitude part and go look at the scalar part and go hey this is not positive. Flip the sign of all of them and now that's the epsilon I switched. So even though I can continuously describe any attitude. In controls, we don't often like this. It's called unwinding problem. I would like it to always go back the short way. Anyway, we going to see a lot of this, I just want to throughout different description so you have been exposed to it. And there are different ways to write, is the same essential equation it's launched in different occasions to different matrix forms. And depending on the application we'll see all of it, so [INAUDIBLE] Good So, that wraps up all the parameters.