0:16

Let me learn some names, so grey shirt.

Â Whatâ€™s your name again?

Â >> Nick. >> Nick,

Â whatâ€™s the difference between Euler parameters and Quaternions?

Â >> They're the same.

Â >> Yeah, basically array of different ways, different theories.

Â But basically, they give you the same thing.

Â There is a field called quaternion math, it's like complex numbers.

Â Instead of just real and one complex, you have real and three complexes.

Â There, the quaternions can have non-unit length.

Â So sometimes you see people talk on their literature about unit quaternions,

Â they're trying to be extra specific for the audience.

Â If we talk to an attitude persons, talk about quaternions,

Â 99.999% of the time they know exactly mean unit quaternion.

Â All right so again, different fields approach this stuff.

Â You got atomic physics, there are other ones, they all deal with this kind of

Â orientation as well as different notations.

Â So good, quaternions.

Â Let's see.

Â Andrei- >> Yup.

Â >> How many quaternions do we have?

Â >> One, four.

Â 1:18

How many coordinates do we have?

Â >> We have four coordinates.

Â >> Right, and in this class we're going to use the Beta notation, right?

Â So we have Beta zero, Beta one, if I can write, Beta two and Beta three.

Â 1:34

So, Spencer.

Â Of these, with the one, two, threes and the zeros,

Â which one are always the same number?

Â And with different notations, which ones might be different?

Â >> [INAUDIBLE] >> Different notations, yeah.

Â Some people do queueus, for example.

Â >> [INAUDIBLE] >> Exactly, so always look for that.

Â If you're dealing with quaternions, it's a fact of life.

Â There's huge different literature, different fields.

Â The parts that are always the same are the vectorial part and

Â that's what we're calling here.

Â In fact, on the last slide, I'm going to call that the epsilon vector.

Â So sometimes, in the control, the beta 1, 2, 3 or the q 1, 2, 3,

Â that's the vectorial part.

Â Now, Robert, why would we call that the vectorial part of the quaternion?

Â Where do you think that came from?

Â 2:27

If you think of the basic definition of quaternions.

Â >> Use the rotation axis [INAUDIBLE]

Â versus to calculate [INAUDIBLE] >> Exactly.

Â E one, two, three which comes from the E hat vector, all right.

Â And this could be in B or N frame if it's the B relative to N attitude.

Â This has, this relates, I mean this is influenced by these vector components,

Â so it matters about which axis have rotated ten degrees, alright?

Â And the rest of it was sign phi over two and that's the same.

Â This one is called the scaler part of the quaternion unit and

Â it only cares about how far you have rotated.

Â That was a benefit of the principal rotation parameter,

Â also a four parameter set.

Â But one of them was explicitly a scalar that only cared about how far,

Â and that's very handy for control applications.

Â I can specify, I got within one degree.

Â I often don't care about which access.

Â I just need to know how tight was my cone.

Â With quaternions you do the same thing, but

Â you just have to look at the beta not part.

Â That's your scalar part all right?

Â So there's that slight distinction that we have here, excellent.

Â 4:10

>> No. >> Okay, If not unirque,

Â how many sets are there?

Â >> Two? >> And then Kevin, right?

Â How are those fifth from MRPs mathematically related to eavch other

Â [INAUDIBLE] >> So right, negative sign.

Â It's a really easy mathematical mapping if you have one set and

Â you want to consider the other set, we just flip the sign, that's it.

Â So Mariel, what's the difference between, you know geometrically speaking what's

Â the difference between one EP parameter description and the other.

Â [INAUDIBLE] Yeah.

Â What does it geometrically mean, having-

Â >> [INAUDIBLE] >> Not quite.

Â You're on the right track.

Â Let's look at the zero, Bill, what is the zero rotation in all the parameters?

Â 4:58

>> As can't written down upside down I'm not sure what you'd say.

Â >> You've got the formula right there.

Â Cosine of 0 is?

Â >> 1. >> There you go, right?

Â 1000, sines of 0 will all be 0, 1000.

Â That's one possible answer.

Â Nathan, what was the other quaternion set?

Â 5:22

>> Okay, what's the different between 1000 and minus 1000 ck?

Â [INAUDIBLE] >> Okay, that's

Â the interpretation of the constraints surface that you're talking about.

Â Good we're about to get into that.

Â But there's a fundamental difference those two descriptions of this orientation.

Â Mandal?

Â >> [INAUDIBLE] >> Right,

Â there's a short and a long, that's it.

Â So this description now gives us the option of describing a short way around or

Â the long way.

Â With the DCM we didn't have that.

Â The DCM was unique.

Â For an attitude, that's only one DCM.

Â That's it.

Â Which is actually very nice.

Â But that's it.

Â 6:11

Principal rotation parameters, there were four possible sets and

Â there was always a short rotation or you could go the long way around, or

Â you could spin about the opposite access, right.

Â Now we don't have to access the options here in all the Euler parameters.

Â But we do have always a short description and the long description.

Â This impacts you when you use them for feedback control because you don't want to

Â feedback an error of 359 degrees, you would rather do minus one, right.

Â Just nodge to your left don't spin the way around.

Â And it impacts in other ways but both sets completely non-singular.

Â You will find differential kinematic equations we derived.

Â 6:49

Our good for either set, it does no mathematical distinction between what's

Â one and what's the other, right?

Â because what happens as I do to Andrew, let say we do a multi-revolution thing,

Â you start out here, this is 1000, you do a revolution, now, you have a minus 1000.

Â And if I do another revolution, I've done two revolutions now,

Â what's my altitude again?

Â >> [INAUDIBLE] >> Now you're back to one, right?

Â So that's why you could keep track with Euler parameters if you've done

Â one revolution but I cannot mathematically distinguish if I've done five or six or

Â seven revolutions right and then the descriptions repeat.

Â You just oscillate the two between, two possible options so

Â you could only keep track of one revolution if you want to.

Â So that's a big part of this as well.

Â Okay, so they are definitely not unique.

Â We have two possible sets, they're non singular.

Â They always work and easy mapping between one set and another is

Â this attitude is equivalent to this attitude.

Â It's just a sign flip.

Â How simple is that?

Â That's cool.

Â Now so in a control, let's say, you integrate these and

Â at some point, you want to switch.

Â Let's say, it's tumbled past 180 degrees.

Â I don't want to describe an error of 200 degrees.

Â I would rather do the short rotation.

Â We could use this math and get there.

Â How, sorry, last row, what was your name?

Â >> Brett. >> Brett, how,

Â if you look at these, Euler parameters,

Â how do you distinguish if you're doing the short rotation or the long rotation?

Â 8:24

>> Beta zero? >> Beta zero, yes.

Â The scaler one Coz if you 280 degrees 180 over

Â two is 90 cosine of 90 is zero, right?

Â So we start out of one, one Beta Naught of one,

Â means I have no rotation then I'm upside down Beta Naught goes to zero and

Â if I complete a revolution it goes to minus one eventually, right?

Â So good we got all of that going, that's the one there.

Â I mean it's mentioned upside down.

Â Now CK you were talking about this hyper surface.

Â Tell me more about that, that sounds interesting.

Â 8:58

>> Okay so it's, it has to be we have a constraint on

Â the ought to have >> Be more precise.

Â Right we have to square.

Â >> The square against.

Â >> Yes, this morning had a question about this, so

Â let me just be precise here quickly.

Â We often say adding up to one.

Â It's implied we mean the norm squared, do you have to square each and

Â add them up right?

Â But that's the mathematical script, no wait that's a two not a three.

Â 9:26

Beta two squared, Beta three squared equals 21.

Â This is your constraint.

Â All right.

Â THis is a four parameter system.

Â It's always a three degree of freedom problem.

Â So we have one constraint.

Â But this means now also, when you integrate these differential equations I

Â showed you last time, when we had the Beta dot equal to this B matrix times Omega.

Â If you integrate this, every time step we tend to reset them,

Â cause they'll be off by just 10 to the minus 13 or something really

Â small depending on your integration steps, but you have to re-normalize them again.

Â With DCMs thereâ€™s also a process of how to reorthogonalize a matrix,

Â itâ€™s just a little more involved.

Â With Euler Parameters, pretty straight forward.

Â Take the, just divide by the norm and

Â you get it back to a normal length to within numerical precision.

Â Good, so we had to apply that there but also deriving and as you said,

Â this really describes a surface.

Â A hyper surface where if you're on one point there is another point on

Â the opposite side that describe the same orientation.

Â So, as you move the trajectory around, you really, your MRP has to evolve in this

Â unit surface and the anti point is the opposite set which can be a shorter

Â [INAUDIBLE] depending on how you rotate it that kind of becomes interchangable.

Â Good, I'll go there first.

Â Okay, Elmer Spencer.

Â >> [INAUDIBLE] >> Yes.

Â >> Is there any way [INAUDIBLE] >> This part.

Â 11:04

You look at the coordinates.

Â So it's kind of you think about it as XYZ space.

Â In that case it would be the x coordinates.

Â If the X coordinates is positive I am on the side has the short rotation.

Â Everything else, the other have the hemisphere has the long rotation.

Â Itâ€™s just hard to think of that in 4D space.

Â But itâ€™s mathematically equivalent.

Â Louise, yeah?

Â Okay. >> If you're integrating those and

Â trying to [INAUDIBLE] strength and if you're doing something like RK

Â 4 you would you enforce that at the end of every time step or

Â at the end of every [INAUDIBLE] >> Good question

Â because we're going to get into this.

Â That's a subtlety but that's a really good question.

Â Let me take that off for now.

Â So the question is essentially, you've got your current set of Betas.

Â All right and then you're using an R K four,

Â how am I going to do that symbolically?

Â It computes a K one sets that's a function of the current stuff,

Â and then you have a K two, a K three, if I can write, a K four, all right.

Â You blend them together and then the final answer ends up being these things summed.

Â 12:42

And you might have really small math errors that build up in between.

Â If you wanted to, you can try and doing there but

Â you'd have to have very large time steps.

Â That is what actually matter.

Â So the renormalizing and just make sure this is always a valid unit for

Â quaterion constraint.

Â Nothing too crazy would happen here.

Â But this is a good question, because as we go to other parameters,

Â especially the modified Rodrigues parameters, this thing will matter.

Â But yeah, practically, I'd go to the next time step,

Â integrate there, then I'd renormalize, and that seems to work pretty well.

Â If you did research and you're really picky, am I doing it right?

Â There's different methods for integrations that you can check.

Â You can check energies,

Â you can check all kinds of stuff to make sure what's my numerical error and

Â what time steps and do these different local discrete approximations make sense.

Â Good, okay.

Â So that was a quick overview of quaternions,

Â they're built on principle rotation parameters.

Â And we went through all those sets including the differential kinematic

Â equation and highlight here, this is something you drive in the homework spot

Â in the end, if I buffer the usual three by one with a zero upfront making it four by

Â one, the only reason we do that is this matrix happens to be orthogonal, right.

Â That was nice, then you can easily invert it.

Â You can re-write this needed.

Â Wait I need to go.

Â There you go, you can go into this form.

Â There's different ways to write this.

Â Typically it's the same.

Â This is now a 4 by 3 matrix.

Â There are some properties here, that are useful.

Â You can easily, this will take two minutes just to double check yourself.

Â But we'll use them later on.

Â Just wanted to highlight them here.

Â And again, sometimes you see the quaternion math written out as what's

Â the differential equation for the scalar part of quaternion and

Â what's the differential equation for the vector part of a quaternion.

Â For example, let's just ahead.

Â The control, let's make sure this actually makes sense.

Â In the controls we typically actually only use the vector part of the quaternion.

Â 15:09

>> I don't know, I would say no.

Â >> Well no, it's 50/50 right.

Â But letâ€™s talk it through, what happens if this part goes to zero?

Â What must a forth or the paramater be in that case?

Â >> Four. >> Right.

Â And it's that the zero orientation.

Â >> [INAUDIBLE] >> Yeah.

Â Negative one is it also a zero orientation.

Â 15:31

It's the same orientation but is it the same path to this orientation?

Â No.

Â And that's where we're going to get in to also now with Rodrigues parameters

Â starting today, wrapping up on Tuesday.

Â Often some, the path to this orientation matter.

Â So if you just go out, so one of them if you just in a control drive this to zero,

Â without looking at what quaternions set you're using,

Â you might rotate all the way around.

Â And you get there, perfect stability, it is not a very nice performing control,

Â there's a much faster path to get there, right.

Â So even in quaternions sometimes people switch.

Â They look at the description and the attitude part and

Â go look at the scalar part and go hey this is not positive.

Â Flip the sign of all of them and now that's the epsilon I switched.

Â So even though I can continuously describe any attitude.

Â In controls, we don't often like this.

Â It's called unwinding problem.

Â I would like it to always go back the short way.

Â Anyway, we going to see a lot of this, I just want to throughout different

Â description so you have been exposed to it.

Â And there are different ways to write, is the same essential equation it's launched

Â in different occasions to different matrix forms.

Â And depending on the application we'll see all of it, so

Â [INAUDIBLE] Good So, that wraps up all the parameters.

Â