0:12

Everybody thinks they know Euler angles, they're like,

Â [SOUND] I've seen that before.

Â Great.

Â So tell me, you look perky back there, back row.

Â Okay, grey shirt, what was your name?

Â Brief.

Â >> Maurice.

Â >> Me or him? >> No, no, back.

Â >> Right here.

Â Thank you. >> What are Euler angles?

Â >> [SOUND] I was one of the people that did not say I have seen those before.

Â No.

Â >> [LAUGH] >> My understanding is, I mean,

Â what comes to mind is rotation about more or less, a [INAUDIBLE] coordinate frame.

Â >> Okay. >> Theta.

Â >> You've definitely heard of them before.

Â Yes. Good.

Â 1:09

>> Right ascension inclination and argument?

Â >> Yeah.

Â If you have orbits, they don't typically call them Euler angles.

Â They just call them ephemeris.

Â These are orbit angles and so forth.

Â But in fact they are Euler angles.

Â And instead of doing ephemeri in, those are three one three sets actually,

Â molar angles, you'll see papers where people use quaternions

Â to define the orbit plane, or MRPs to define orbit planes.

Â And there's whole other ways of defining these orientations of the plane.

Â So you can see all these worlds kind of cross couple like that.

Â Good.

Â But basically, as we say, something like yaw, pitch and roll.

Â And again, NASA was the big culprit there for awhile,

Â claiming that they're easy to visualize.

Â 2:11

>> Yeah, so I already forgot the numbers you said.

Â >> Three, two, one.

Â 3 degrees yaw, 2 degrees pitch, 1 degree roll.

Â >> Yes so, yaw, and then pitch and- >> Okay.

Â So you went through the sequence of motions, that's good.

Â because that is related to what you were talking about.

Â Euler angles are actually a sequential rotation sequence.

Â 2:34

So this is the mathematics, we are adding.

Â You're doing, basically your first ration about one of these orthogonal base vectors

Â that Maurice was talking about.

Â So you have your right hand coordinate frame, B and N are equivalent, and

Â you're saying, I'm doing my 3 degrees yaw, then I'm doing my 2 degrees pitch.

Â And 1 degree rolled and to think about it for a second.

Â So if you have your thumps up like this, pitch is actually down,

Â you put your thumb along that axis curl your fingers in that case it's down.

Â That's why often shifts to find it differently so pitch on a ship is up.

Â So just be careful how those frames are defined on how we do that or

Â on an aircraft, we define it differently.

Â Where is your yaw access on an aircraft?

Â 3:19

Down actually and where's your roll access on an aircraft?

Â >> Right along the.

Â >> Yeah, along the forefront, that's your pitch axis that you're thinking of.

Â So the one axis go forward,

Â the two axis goes to the right wing, three axis goes down.

Â That's why a positive yaw is to the right, a positive pitch is actually pitching up.

Â But a positive roll means you're tilting to the right.

Â That's how they define the frame.

Â 3:45

So I'm using these numbers, it's a sequential rotation sequence.

Â Yes?

Â >> Just say positive pitches down.

Â >> That's how you define your thing.

Â If I have it this way, positive pitches down.

Â If I define my frame this way, then a positive pitch is up,

Â because one has to pitch to the left, one has to pitch to the right.

Â So be careful how that's actually defined.

Â 4:19

How many sets of Euler angles are there?

Â >> 12.

Â >> 12 actually.

Â And we're going to look at that.

Â And what I want you to do is, I want you to be experts in all 12 sets.

Â Now, you don't have to memorize 12 sets of equations and transformations and

Â all of that stuff.

Â But there's clear categorizations of those 12 sets of all the angles,

Â this grouping behaves this way this grouping behaves this way.

Â These are the singularities there's three parameters that must be singularities.

Â What are the differential kinematic equations?

Â How do we add them?

Â How do we subtract them?

Â They're not vectors.

Â Even though we write yaw pitch role often as a three by one set,

Â it's not a vector set.

Â We can't just advert.

Â So we're going to look at all those different things.

Â We'll start today.

Â We're not going to finish today.

Â We'll going to be wrapping up, specially the additions and that sort of stuff,

Â the next lecture.

Â So really, they're very common set and while you hear me

Â making maybe at times comments that might sound disparaging like the total crap.

Â People might take that negatively.

Â They actually do have big values especially in robotic systems.

Â You often have shoulder links that, this link can move this way.

Â And then, your arm can twist this way and the last one can do other motion.

Â And the often mechanical building systems that are sequential rotation sequences

Â where this math is absolutely perfect.

Â Where I'm less thrilled with them is three dimensional rotations.

Â because you'll see they're the most singular set that we have.

Â You're never further than 90 degrees away from a singularity.

Â 5:50

There's much better sets to use, general spacecraft rotations.

Â But they're a very fundamental set.

Â For small angles, people are very familiar with yaw, pitch, roll,

Â kind of aircraft-like things.

Â So you definitely, this is one of your tools in your Swiss Army knife, and

Â you really need to know what they are.

Â Even though it may not be a primary tool for spacecraft dynamics,

Â you have to understand them and how they come.

Â So very common set. The other key is sequential so

Â when Evan was doing his motion.

Â Really this is where you want to use your right hand again not your left hand.

Â because everything is right-handed here.

Â You start out with the original orientation.

Â And now we're going to specify how we are going to rotate and

Â the order actually matters.

Â If we had done the roll, and then the pitch, and then the yaw,

Â you get a different attitude, and I'll illustrate this in a second.

Â So that's why the sequence, I always love this when I get these angles and,

Â here's your and they give me something like this is your roll, pitch, yaw.

Â Which immediately raises flag for me.

Â Why are you saying roll, pitch, yaw and not yaw, pitch, roll?

Â So I asked him and emailed him and said, this is a three, two, one sequence.

Â I get back the question mark.

Â What do you mean?

Â I'm like, aw, crap.

Â We're in trouble.

Â Just give me the math.

Â If I look at the math I know precisely what you're doing.

Â And some people do do a one two three sequence.

Â Others have a three, two, one sequence.

Â They're different.

Â They may linearize the equivalent but for

Â large [INAUDIBLE] anything beyond the linear regime, they're not the same thing.

Â So we really want to be careful.

Â The way we label them, you see here, I have an IJK.

Â So this could be three, two, one.

Â The orbit was a three, one, three.

Â And there's different combinations.

Â In fact, we said 12.

Â For the first axis, let's go through that.

Â As Maurice was saying we have to, with the all angle sequence we're not rotating

Â about some arm but through body axis.

Â We're using a primary one, two and three.

Â That's why when you define your body-fixed frame this is critical.

Â What is one?

Â What is two?

Â What is three?

Â Again, space station has lots of body-fixed frames.

Â So you have to be careful.

Â Now, how many options do you have for your first choice, first rotation?

Â 8:07

Why two?

Â >> You would a two axis rotation,

Â another two axis rotation would be the same as just adding two-

Â >> Exactly, because then you introduce,

Â you don't get three degrees of freedom,

Â you've reused the same degree of freedom twice, sequentially.

Â That's not giving you any new stuff.

Â So if you did a two, you can't do two again, but I can do one or three.

Â So let's pick one, now I get a two, one.

Â Now, that I did the one rotation, how many options do I have for the last one?

Â Two again.

Â I could reuse two again, or I can go three.

Â If I want to spread out across all the axis.

Â And that would be some of the fundamental differences.

Â But it's 3 times 2 times 2, which is 12.

Â That's how we come up with 12 possible combinations of all the angles.

Â 8:50

Okay, good.

Â Easy to visualize for small rotations.

Â Again, I can just give you this angles and you can quickly go.

Â Do the quick twists with your wrists and that must be the attitude.

Â That's kind of the nice thing about them, somewhat intuitive.

Â And it's good to know.

Â If you have large rotations, and you'll see some animations, then it's trickier,

Â if I tell you hey your yaws 180, your inclination is 1,

Â and your pitch is minus 140 what's the attitude?

Â And I really have to go through the complete sequence.

Â I can't visualize quickly, is that large a departure or not?

Â because with all our angles, you can have stuff where you move far,

Â you twitch a little bit, and you almost move all the way back.

Â Now, you have two very large angles but

Â actually the angular difference between two frames could be very very small.

Â That's some of the stuff that you'll see.

Â So here's my quick spacecraft thing that I drew up, well, 30 years ago.

Â But this is like an aircraft like frame.

Â We talked about the third axis points down.

Â First axis points towards the nose and

Â the second axis points towards the right wing tip.

Â This way you put your thumb out there.

Â That's a positive pitch up, positive yaw to the right and

Â a positive roll has you dipping to the right.

Â But this not universal, ships actually have the yaw typically up.

Â So their pitch warps, and it's different.

Â 10:22

The sequence.

Â This is just a mathematical illustration.

Â So you really go from, originally if you have 40 degrees, 60 degrees,

Â minus 50, I would have to go, I'm going to rotate 40 degrees,

Â then I'm going to do a 3 2, that's this 1.

Â What did I say?

Â 50, 40, that would be down.

Â And then, the last one, negative this way, and

Â that would be roughly the final attitude.

Â But it's a sequence of rotations.

Â And as we go through this I used these notations a little bit in my development.

Â As we're doing a sequence, I'm starting out with b and n being identical.

Â Then, I do my first rotation.

Â This really creates a new frame.

Â That's what I call the b prime frame.

Â Then I do the second rotation, and then that gets me the double prime frame.

Â And then, I do the final rotation, which could give you the triple prime.

Â But really, the final rotation is the body orientation.

Â That is the spacecraft relative to this inertial frame.

Â But you went through some sequence to get there.

Â So if you see primes and double primes,

Â those are those intermediate first two frames, and it'll be useful when

Â we develop the differential kinematic equations to relate these rotation axises.

Â 11:32

Okay, good.

Â This is, said that was a 3-2-1 sequence we looked at, which is what we have for

Â aircraft also sometimes a spacecraft.

Â Orbits has this one to 3-1-3 and really you do your first

Â a 3 rotation that's about your north pole axis here and

Â that goes up and that gives you your line of nodes.

Â Then, we do a one takes a one axis from here to here the one axis now kind of like

Â one prime, becomes your Axis about which you're doing inclination.

Â And then now you're in the orbit plane.

Â Your third axis now normal to the orbit plane, and we rotate about the b3 to

Â switch up our line of periapsis up to here, it's a 3-1-3 sequence.

Â So for a 3-1-3 sequence, where is the singularity going to occur?

Â 12:50

Alright so we do a three-one-three, two-one-two,

Â one-two-one, three-one-three, and so forth, that's what we have.

Â Then, always this angle is zero.

Â That's one definition for geometric interpretation for singularity.

Â What's the other singularity we have?

Â 180, right?

Â In fact, if you flip it.

Â So that's why you typically see in orbits,

Â your angles are always defined between 0 and 180.

Â Otherwise the ascending node is defined between 0 and 360, 0 and 360.

Â So they move the nodes around

Â such that your inclinations are always a positive value, never a negative value.

Â 13:29

So that kind of clips the range of these coordinates.

Â You never have minus 10 inclination.

Â Okay, good dude, this one's easier to visualize.

Â Anybody remember three to one?

Â Where there are singularities happen?

Â >> It should 90 degrees?

Â >> Yes, second angle and then it's 90, plus or minus.

Â Again, there's two possible ones and you will see the mathematics here shortly.

Â But it's true actually for all of the asymmetric sets.

Â One, two, three.

Â Three, two, one.

Â Two, one, three.

Â Any of those.

Â It's always going to be the second angle, and plus minus 90.

Â Whatever's going on in the math.

Â That's where it goes singular.

Â So those are some nice patterns.

Â Here's some quick videos we can see.

Â I'm showing you essentially a three, two, one sequence of 60,

Â 50, 70, and a three, one, three of 60, 50, 70, just showing that the order matters.

Â If you rotate by different axis, you get different amounts of final attitudes.

Â So not too surprising.

Â But If I keep the same rotation on the left hand side and

Â the right hand side, I adjust me three-one-three angles to be this.

Â 14:41

So in essence this means, how do I convert 60,50,70,

Â 3-2-1 Euler Angels into the equivalent 3-1-3?

Â This is an essence according to transformation that you have to do.

Â As like saying, look,

Â I've gotta cartesian position and then now I have to have equivalent spherical.

Â We have to be able to translate between all of this different And I'll show you.

Â In fact, the easiest way to do this is through the direction cosine matrix, and

Â we'll show that in a second, how we relate those to each other.

Â But this is just a visualization that you do get there, but

Â it's different rotation sequences that we do.

Â 15:16

Here's another one, where I have a one, three,

Â two sequence And you can see it has two larger angles and one really small angle.

Â And that gets you there so again to me this highlights how misleading sometimes

Â it can be if you have these big angles.

Â Does that mean you're really far away from where you want to be?

Â And its not always the case with Euler angles you can move over here,

Â do a little twitch almost come back And in here in this case,

Â there's two larger angles, but one of them is almost zero.

Â So that kind of immediately tells me that it's actually not that far

Â away from it again.

Â 16:46

For a symmetric set, and I'm using the 3-1-3 as a representation of that.

Â It's just easier to visualize and people familiar with the orbits.

Â It's if the Planation\g is 0, 180.

Â Then you repeat rotations about the n3 or minus n3.

Â Either way, n3, minus n3, it's still the same axis about which you're rotating.

Â And that's where you lose the uniqueness issues.

Â If it's an asymmetric set, it's plus, minus 90 degrees,

Â which is basically pitch 90 up and down.

Â A little bit harder to visualize.

Â But there, too,

Â you end up having mathematical ambiguities in your mathematics.

Â Of describing it and extracting these angles.

Â You can see with zero to 180, the best you can be with 90 degrees, a polar orbit.

Â And now you're 90 degrees away from a singularity.

Â That's as far as it goes.

Â With these, the pitch, you'd be at zero pitch.

Â That puts you right in the sweet spot between the singular behavior.

Â But you're never more than 90 degrees away.

Â Just keep that in mind as we look at other attitude sets, and start to compare.

Â You got a question?

Â >> No.

Â >> No. You're just doing your finger things.

Â Perfect.

Â Following along.

Â Good.

Â These are the singularities we discussed.

Â