0:18

But they're very, very popular in spacecraft, or they have been.

Â These days, I would say, modified with rigorous parameters is starting to

Â make a pretty good run for it as well for this title.

Â They're becoming quite popular.

Â Different coordinate sets people are coming up with as well,

Â depends on the application.

Â But why do we call them Euler parameters?

Â Or sometimes, they call them quaternions?

Â Depends on your history, how this is done.

Â Different papers have different notations.

Â The way I was raised, they always called them Euler parameters.

Â These days, they call them more often quaternions.

Â I don't know why, but they're basically interchangeable.

Â If you see one name, if it's Euler parameters, and

Â that's what the book calls them too, we tend to use betas.

Â That's the common notation you'll find in the literature for that.

Â If you have quaternions, people like Q, so, it's all little q's.

Â 1:02

But there's still [LAUGH] questions even there with quaternions.

Â Again, notations are wonderful here.

Â They're always different.

Â There's four quaternions that we have in the end.

Â And some people tend to write these quaternions as q1, q2, q3 and q4.

Â And some people write quaternions as q0, q1, q2, q3.

Â So, if you're reading a paper, different textbook.

Â If I'm dealing with quaternions and q's, the first thing I always look for

Â is, where is that off one?

Â These three, q1, 2 and 3, will always be the same.

Â I haven't seen those interchanged.

Â Nobody's been quite that cruel yet.

Â So, please don't go there.

Â But the q0 or q4, the fourth one is either on the fourth slot or

Â the zero slot, notationally.

Â So, it just depends which text, which training they had,

Â that's where they put it.

Â 2:13

So, all kinds of history there.

Â It's a four-parameter set that we're dealing with here.

Â So, let's talk about this a little bit.

Â They're very popular for

Â spacecraft because they don't have singularities, that's one of them.

Â But we've already seen an attitude of scope that doesn't have singularities.

Â Chuck, which other attitude set that we've discussed so

Â far doesn't have any singularities?

Â >> A DCM? >> A DCM, exactly.

Â With the DCM, what were some of challenges, Warda, Warda.

Â Warda?

Â Yes, what were some of the challenge of the DCMs, of dealing with that?

Â >> A lot of computation is required.

Â >> Yeah, so, it's a 3 by 3 times a 3 by 3, it's a lot of stuff.

Â Now, it's linear math. Actually,

Â computers are very fast with that.

Â But the big challenge is how many constraints do these coordinates have?

Â because we got nine coordinates.

Â Six, right?

Â So, when you integrate, you have to be very careful at every integration time

Â step that you still satisfy all these constraints.

Â Otherwise, your description is going to blow up and you end up with a matrix,

Â kind of what you were talking about earlier, Robert.

Â Where the columns are not unit length, the rows aren't unit length.

Â And so, you always have to reapply those constraints, and that's a challenge.

Â Now, here, we're dealing as a four-parameter set.

Â So, how many constraints are we going to have here?

Â One, right?

Â So, only 1, so, that's a nice reduction.

Â There are other benefits.

Â I'm giving you quick preview.

Â There's some really simple math.

Â The differential kinematic equation was pretty nice for the DCMs.

Â It turns out it's really nice for this, all the parameters as well.

Â And that's, it's this bilinear form.

Â It has good things if you take estimation theory, and have to do common filters.

Â And you have to linearize this stuff.

Â Well, and these things are already in bilinear form.

Â It's really easy and

Â you don't have to lose any accuracy just because you need a linear form.

Â So, there's some really nice addition properties as well that we'll look at and

Â in fact, you get to derive.

Â That's always a highlight in this class.

Â You can really see how good you are with algebra.

Â 4:15

To do that, but it's an elegant property and

Â you will own it by the end of this stuff.

Â It's a benefit, it's non-singular, linear differential kinematic equation.

Â It works really for any small orientations, large orientations,

Â challenges, constraints.

Â There is a constraint, but only one,

Â compared to the DCM where we had six essentially to apply.

Â And then, you'll see, it's very easy to reapply this constraint.

Â Not as simple to visualize, true, but we can't quite just use the fingers and

Â do the sequences.

Â But as we work with these parameters, I hope you get a little bit more intuition.

Â I'm going to keep throwing out tips and tricks about this.

Â This is how I can look at quaternions and quickly tell that's roughly pointing here.

Â Or if I use it for a control problem, am I tracking well or not?

Â What should I expect?

Â Right, there's some nice patterns.

Â It works pretty well to go there.

Â 5:11

But you could also replace q0, 1, 2, 3.

Â Then, we use everything in terms of beta.

Â Beta 0, 1, 2, 3.

Â This is how they're defined.

Â And they're defined now directly in terms of the principal

Â rotation angle and principal rotation access vector components.

Â So, Shayla, these e1, 2, 3's.

Â If this describes the orientation from n to b,

Â are these vector components taken in the n frame or the b frame?

Â 6:05

It was an eigenvector.

Â So, therefore, the DCM times e-hat gives you back the same DCM.

Â So, all these little tips and tricks to help remember this.

Â Good, so, I'll keep asking Pingy on this.

Â Hope everytime we go through, it will sink in a little bit more.

Â So, e1, 2, and 3, I'm not saying explicitly this is b or n frame,

Â it could be either.

Â What I do know it's not q frame or r frame, right?

Â This is the attitude from n to b, so, either it's b or n, one of the two.

Â That's the short hand.

Â So, now, with this stuff, you can see the 1, 2,

Â 3 components directly relate to the vector components, 1, 2, and 3.

Â In fact, it's the vector components scaled times sine of the half angle.

Â So, the other thing is to see, we don't deal with fee directly.

Â It's always half fee, half angles.

Â And that appears everywhere in the math.

Â 6:54

So, some triggered entities, you'll have to use the half or

Â doubled-angled identities to get these stuff to work out.

Â So, that's kind of what they're defined.

Â So, therefore, if I did a pure 180-degree rotation about my 3 axis.

Â In that case, e1 is going to be 0, e2 is 0, e3 is 1, right?

Â It's just 0, 0, 1.

Â That's my B3 axis.

Â So, 0, 0, this is 1.

Â I'm doing 180 divided by 2 is 90.

Â So, B3 in that case is one.

Â And these are all going to be zero.

Â And if you want to look at this other one here.

Â 180 over 2 is 90.

Â Cosine of 90 is also 0.

Â Because this is kind of how you apply it there.

Â So the one, two, threes are the ones that are proportional to vector components

Â times sign of half the angle.

Â But this is the part that I typically remember,the E one,two,three.So

Â if I know I am doing a pure rotation about one axis then I will only have

Â a beta one,a beta two or beta three right,everything else will go to zero.This

Â is the one we often use to tell us how close are we.

Â 8:01

So similar as with the principal rotation parameters to have a niat at an angle.

Â And the angle is just one angle and

Â it tells me hey, your estimation is good within two degrees.

Â I don't care about which axis, it's just good within two degrees.

Â At the same time, I can look at this one.

Â There's no axis, it doesn't care about which axis have gone to go from N to B.

Â It just tells me how much was the principal rotation angle divided by 2,

Â cosine of that, that's it.

Â So it's a single scalar measurement.

Â We just don't have quite the direct geometric interpretation of the principal

Â rotation angle, but it's still a single scalar part.

Â So let's look at it again.

Â If we have b and n are exactly the same attitude.

Â 8:45

In this case, what is the principle rotation angle?

Â 0, right?

Â E-hat in this case is not defined, but

Â the principle rotation angle is very well defined.

Â So the ambiguity just shows up in this, but phi is 0, so

Â 0 times whatever finite you picked is going to be 0.

Â Cosine of zero just gives you one.

Â So the zero rotation

Â ends up just being one on what's called the scalar part of the quaternion.

Â This is often referred to as the vectorial part of the quaternion because it

Â depends on the e hat vector.

Â The scalar part does not depend on e hat vector, just the angle.

Â So lots of little definitions.

Â So how do we now interpret this?

Â Zero rotation, I have one zero zero zero.

Â That's it, now I'm there.

Â 9:37

What is 360 degrees principal rotation axis, Trevor?

Â How does that differ in my attitude,

Â how does bnn relate if I have a three hundred and sixty degree rotation?

Â >> It's the same >> It's the same, so

Â if I have 360 degrees and divide by two,

Â take the cosine, what do you get?

Â >> You get one.

Â 10:19

Now doesn't that strike you a little bit odd?

Â We have exactly the same orientation.

Â Up to now, well at least for the DCM's and the Euler angles if you pick

Â an orientation another orientation and compute these things, I only get one DCM.

Â It's non singular, and it was unique.

Â Euler angles the same thing.

Â I only get one set of Euler angles, because they're bounded by those,

Â you know, four quadrants or the two quadrants depending on the set.

Â That's it, principle rotation parameters all of a sudden,

Â you get four possible sets.

Â Here also, the quaternions are not unique because the same orientation

Â gives me now two different sets.

Â So what's the answer?

Â If I'm telling you b and n are identical,

Â do you pick beta not equal to 1 or minus 1?

Â 11:25

>> Because if I'm just given only the fact that b and n are the same and

Â I assume that there is no rotation.

Â >> Right, it's a common thing, right?

Â Depends on your applications too.

Â The only difference between beta not being one and

Â beta not being minus one is the path at which you arrive at this rotation.

Â One of them says you've done at least one revolution, all right?

Â The other one says you're just there.

Â 12:07

So that's implied in this.

Â If you have a tumbling body which in spacecraft we often have.

Â You've deployed off a rocket, you kicked off, it's tumbling, and

Â you just integrate your betas, they will go from 1 to -1.

Â The beta 0 would go from 1 to -1 to 1 to -1.

Â And every time you go to 1, you can't say, well,

Â now I have unwound my spacecraft, right.

Â All you can say if you've gone back to the original history.

Â But at some point, you have to start and say, that's my reference.

Â Time to 0, this is what I've done.

Â And you'd have to count such occurences, all right.

Â So it doesn't, it just gives you, I haven't done, I've done at least one.

Â And if you think beyond,

Â I would have to keep track in my code to see if I'm tumbled six times.

Â The proturnians won't tell you if you've tumbled six times.

Â It'll only tell you if you've tumbled at least once.

Â This is only actually two sets in these things.

Â Okay, good so the zero rotation for

Â other angles becomes just all zero Euler angles typically.

Â We have this rotation angle just goes to zero for the zero rotation.

Â That's fine, DCM, what's the zero rotation for the DCM?

Â 13:28

So, different ways to interpret this stuff.

Â If you've tumbled, the worst attitude error you can have is 180 degrees and

Â it doesn't matter about which axis I do it.

Â If I have 180 degrees divided by two, is 90.

Â This cosine goes to 0.

Â That means I've done 180 degrees.

Â So if this one goes to 0, I've now tumbled upside down.

Â 14:05

I'm going from 180 point Trevor was talking about onward to 360.

Â So, now I'm just describing the long rotation.

Â So, there's another one thing, is that I can just,

Â if I'm looking at all for look for the scalar one.

Â And just from this definition, you can see, well, if it's positive,

Â it's describing a short rotation.

Â If it's negative, it's describing a long rotation.

Â And if it's zero, if beta naught goes to zero,

Â I know this craft is pointing upside down relative to whatever the other frame is.

Â All right, so we'll be using this.

Â This will sink in hopefully more, and more, and more as we go through it.

Â But this is the basic definition in terms of principle rotation parameters.

Â Now there's constraints here we have to deal with.

Â The e1, 2, 3s, they come from a unit vector,

Â therefore e1 squared + e2 squared + e3 squared has to equal to 1.

Â When you do a lot of Euler Parameter property validation,

Â derivation, as you do this next homework, you definitely want to use this.

Â Because then also, what comes out of this too is if you take these betas,

Â you square them and add them up, you will end up with cosine squared

Â plus sine squared times factored out e1, e2, e3 squared.

Â Those three sums squared to 1.

Â This just leaves you with cosine squared plus sine squared,

Â which hopefully you remember from basic trig is just 1.

Â So this now means in 3 dimensional space, what type of geometry do we describe?

Â When you have x squared, and y squared, and z squared equal to 1?

Â >> [INAUDIBLE] >> It's a sphere,

Â which part of the sphere, Chuck?

Â 15:45

>> Sure. >> Anywhere around the sphere,

Â inside the sphere, on the sphere, right?

Â It's the surface, but I want to be very precise here.

Â It's the surface, so this really describes the surface

Â of these coordinates have to lie on a surface of a unit sphere.

Â A three dimensional unit sphere.

Â 16:11

>> Hypersphere.

Â >> Yeah so if you had heard hyper in geometry.

Â Basically means we're waving our hands and giving up.

Â It's anything more than three, all right?

Â Could be four, could be five.

Â There's actually descriptions from six dimensional space.

Â You can have six dimensional hyper whatever planes and stuff.

Â What does that mean?

Â It's like a three dimensional thing but in a higher dimensional space.

Â So this resides on a three dimensional surface.

Â Unit distance, so it's a unit sphere in a four dimensional space.

Â So a surface in a four dimensional space is a three dimensional manifold,

Â a three dimensional sub space of that.

Â That's where these things lie.

Â Now who can visualize that?

Â I can't, right?

Â So how I draw it typically is like this.

Â I'm just cheating.

Â I'm just drawing a ball.

Â You know, it's the easiest way to map it actually.

Â And there's a whole papers on how to take geography mapping projections,

Â Mercator mapping, other kind of stuff to map these things into other

Â coordinate sets to so the analogy to the earth globe is pretty good.

Â You see this in lots of attitudes papers so

Â it just means that these parameters have to lie somewhere on this unit surface.

Â This is the concurring constraint when we integrate and

Â the length of this is a unit concurnium.

Â Concurnian math can actually be done for non-unit length as well.

Â That's how who the quaternions?

Â Is that Calvin?

Â Hamilton?

Â Who knows?

Â I think it was another applied mathematician with way too much

Â time on their hand.

Â I should look that up.

Â 17:42

But you can see quaternion math where quaternions don't have to be at length.

Â Quaternion math looks a lot like complex number math.

Â You've got a real one, the scalar, and then these others are like these imaginary

Â axes but instead of one imaginary axis we have three imaginary axes in a scalar.

Â That's how quaternion math was built.

Â And now if you actually make it a unit quaternion it turns out all these

Â wonderful properties come up.

Â I'm not showing you any three dimensional complex numbers stuff here.

Â We're just going straight to the principle rotation parameter interpretation.

Â We want to see that geometry.

Â It's really easy to get to the same conclusion.

Â So think of these having to reside on there.

Â Now it turns out the other quaternion set or the parameter set is simply minus beta.

Â 18:26

We saw that for the zero rotation.

Â It was either one, zero, zero or if it had 360.

Â Which is the long way around to the same orientation, it was minus 1000.

Â So and I'll prove this on the next slide going we had four sets of

Â principle rotation parameters.

Â If you take those and

Â apply those into these definitions, you end up only with two unique kurturnions.

Â So I'll show you on the next slide.

Â But before I go there, let's do a quick vis.

Â Sometimes just 3D geometry trips people up sometimes a little bit.

Â So what we really have is I'm drawing a red ball.

Â That's on our, pretend that this is a four dimensional space.

Â I'm just showing you a three dimensional illustration of that.

Â But any point,

Â any attitude you do has a certain location on this surface, all right?

Â So this point on that surface means 30 degrees about this axis.

Â Another point, that means 65 degrees about a different axis.

Â But every orientation can be described as a point on that surface.

Â 19:28

Now If this is my beta naught axis, if I'm on this half of that surface,

Â beta naught being positive means I'm describing the short rotation.

Â If this point now moves around to the far side, I'm describing the long rotation.

Â So if one quaternion set goes to the long rotation, the other,

Â the alternate quaternion set, has to become the short rotation.

Â In fact, that's what we were talking about earlier.

Â If I have beta not equal to one do I know it's a zero rotation?

Â Well, technically not even.

Â I could have done 720, right.

Â Those two things just keep flip-flopping back and

Â forth as you would continue to tumble and tumble and tumble.

Â But that's essentially it.

Â So anywhere on the surface if you have an attitude.

Â There's always an anti-point that basically describes

Â exactly the same orientation, as Trevor was saying.

Â But, it describes a different path to that orientation.

Â One's a short, one's a long.

Â And it doesn't keep track of multi-revolution,

Â it only keeps track of within one revolution, short and long.

Â That's it.

Â That's what we have.

Â So, Moving on.

Â That's the basic definition, four parameters, one constraint.

Â They are non-singular.

Â This works everywhere but we have to deal with this constraint.

Â