0:21

So, but the resulting motion, just looking at these yaw, pitch, rolls relative to

Â inertial frame, they were all over the place, and very nasty to predict.

Â Lots of wiggles, bumps, and stuff that was going on.

Â So, it wasn't very intuitive to predict what happens to the natural

Â procession of a torque-free attitude motion.

Â So this is a classic result that people have done.

Â 1:10

Who thinks mutation procession definitions are confusing?

Â >> Like slowly around the depth.

Â >> Good. I hate them, quite frankly.

Â I'm always confused by those darn things.

Â Why?

Â In the end, those mutation procession things,

Â as I'll show you, they're nothing but older angle rates.

Â 1:27

And, as often in the literature, we're not consistent.

Â Some people talk about 3-1-3 Euler angle rates which means

Â certain motions mean something.

Â That's the notation part.

Â That's the procession part.

Â It's a 3-1-3 versus a 3-2-1.

Â The processions are often very similar because they are both 3 first axis.

Â But the 3-2-1s and the 3-1-3s are the two common things you will see people

Â use to describe torque free attitude motions and look at the tumbles.

Â But they don't always tell you what they are doing until you look at the math,

Â it takes a long time so I'd rather they just tell me,

Â look I'm studying it using three one three over under rates.

Â I know right away what you are talking about verses mutation and damping.

Â But it is a common mutation, common thing you hear so let's cover it and

Â I want to show you a little bit what it does.

Â The other issue is so we are going to use older angle rates.

Â to start to prescribe how much does this wobbling disk process around and

Â what's happening there.

Â The other trick is, whereas my earlier example acts as a metric,

Â I was plotting my yaw pitch roll angles relative to the inertial frame.

Â With nutation precession,

Â those terms, we're never talking about attitude motion relative to inertial.

Â We're talking about attitude motion relative to your momentum vector.

Â 2:38

That's the key difference.

Â So, if this thing is spinning this way,

Â you have a big momentum vector sticking up.

Â Great!

Â Then you talk about precessing about that momentum vector and

Â how much you're nutating and wobbling Around the momentum.

Â If you're spinning this way,

Â then procession is really defined around that momentum vector, right?

Â So it has nothing to do with where your inertial system is,

Â it's torque-free motion.

Â Momentum vector is preserved.

Â So, if we do that, let's go through this.

Â I know some of you have seen this.

Â So, you have some momentum vector, it's torque-free motion,

Â as seen by inertial observer, this is a fixed quantity.

Â You give something some spin It's going to have some fixed momentum.

Â And whatever else happens and

Â how it gyrates and wobbles, that momentum has to be fixed.

Â So, we can now pick an inertial frame that is lined up with n3.

Â Just historically they pick the third one for some reason, to be = -H.

Â 3:31

So now, I can talk about attitude motion relative to this frame here.

Â And n one two are really arbitrary,

Â they don't really matter much in the end you will find.

Â So I can write the momentum vector as minus h times n three and

Â this is also an frame, right?

Â It's just lined, however you released it, that momentum was this direction.

Â So n three goes in the opposite, that's it.

Â So now we can write that and we can say okay, good this makes my

Â h vector in this particular end frame, all right this is a special end frame,

Â not a general end frame, is zero zero something times a dcm.

Â This allows me to write my momentum

Â vector components in the body frame in an easier way.

Â Generally this H if you have an arbitrary inertial frame H would have an H1 two and

Â three in frame component.

Â Here we only have a third component, so

Â that means times the DCM you are not only pulling out the parts of the third column.

Â That saves you lots of Algebra.

Â 4:32

Why do we do that?

Â So here's the DCM and using 3-2-1 Euler angles here so

Â I'll get 3-2-1 Euler angle rates out of this stuff times that zero zero minus H.

Â You do this, you have the H times sine and the minus minus cancel.

Â Here we have H with a minus and a sine and a cosine of the third angle,

Â which is rolled and the second angle which is pitch.

Â So phi is rolled, theta is pitch, and the same thing for the third term.

Â And this is the momentum vector in the body frame.

Â But then, you remember from kinetics, H is also equal to i omega, right?

Â That's what we derived for rotational motion.

Â And in principle coordinate frame.

Â This is simply i one, omega one, i two, omega two, i three, omega three.

Â So this is an elegant way that we can use the three dimensional and get momentum

Â vector to provide three constraints on how does omega and the angles have to roll it.

Â 5:36

Which kind of makes sense because there will be the motion about the momentum

Â ,vector that happens there.

Â So, if this is the spinning and

Â the processing rate you're going to have, if we take the system and twist it for

Â 80 degrees about the momentum vector, it's completely asymmetric.

Â It gives you the same kind of a wobbly motion.

Â So that's kind of what manifests in the Mathew, does know we are.

Â There is H and these omegas.

Â Well, this is nice, so I don't have to integrate differential kinematic equations

Â to come up with, at this attitude, what is omega going to be?

Â It's kind of like the energy conservation, the bouncing ball problem, right?

Â That we've used the complete three dimensional h vector

Â here instead of just the magnitude as we did with pole plots.

Â So that's good.

Â So now, we can divide by inertia and you get this expression.

Â So, I can write analytically, if this is the pitch and this is the roll,

Â these are the omegas you must have or otherwise you're violating momentum.

Â 6:25

So good, we have that.

Â Now the next step is we have our differential kinematic equations for

Â my Euler angle rates and omega.

Â Now I just had an expression of omegas in terms of these angles and you plug that

Â in so instead of having a differential kinematic for theta, you know,

Â your state rates then you differentiate omega to get the accelerations,

Â here we used angular momentum to completely get rid of omega dots and

Â I get, I end up with torque free motion that's three coupled non linear.

Â But first order of differential equations.

Â So I got, I used angular momentum to get rid of some of those dots.

Â Didn't have double dots in the end, in attitude angles, I only have single dots.

Â And this would be your procession,

Â this would be your mutation if you used a 3-2-1.

Â And then, the third roll, we often just call roll as the final roll motion.

Â But that's the wobbly plate problem, we've all seen this plate on a table, right,

Â that wobbles It doesn't keep the same plane, that plate kind of goes around and

Â precession and does some weird stuff, right?

Â That's all going to be predicted now by these motions.

Â If you look at this, positive, positive, H magnitude is positive, sin squared,

Â cosine squared, all positive.

Â The precession rate, your yaw rate essentially, is always negative.

Â And this comes out of the definition that we've chosen and end frame which is

Â in the minus H direction otherwise there will be a sign flip.

Â So is different this could be different but this is more the classic one

Â whereas the pitching, that's basically a mutation rate.

Â Does this, if this spinning, is there an up and

Â down motion that happens because of this?

Â And you can see, for example, if i3 is equal to i2.

Â This term actually vanishes and you get rid of notation rates, as well.

Â So different shapes call certain rate things to change.

Â 8:18

Anyway, so this is the classic one for general inertias, and

Â we cannot be positive.

Â If you have an axis symmetric case, so

Â I'm just assuming here B1 is my axis of symmetry.

Â So, I1 is unique and I2 and I3 are equal.

Â And you plug it into those equations.

Â They simply fly down to a constant mutation rate and

Â a constant procession rate that you would have.

Â This would tell you that if the sphere is spinning,

Â that at about the momentum vector, this object is spinning, but

Â it's also slowly processing around.

Â It has to do that to conserve angular momentum.

Â And so, these are some classic results where you can now actually make

Â predictions about the attitude motions,

Â and turns out all their angles are kind of convenient for some of this stuff.

Â They're nicely to be worked out.

Â And this tells you what happens to the motion, since data is constant,

Â your roll rate that you would have in this situation also will stay constant.

Â So this is pretty classical.

Â We build a lot of spaceships that are kind of, at least originally,

Â almost cylindrical, very axi-symmetric just because they come out of rockets and

Â it's been released and have a wobble.

Â These equations are very common to kind of analyze what happens post tip off,

Â how are things going to process, what's going to go on.

Â So, it's a convenient form.

Â 9:29

Now, there's some other stuff people.

Â I'm just going to highlight this.

Â We're not going to go through details.

Â There's no particular homeworks on this.

Â But if you have angular momentum vector, here, this is my axis of symmetry, for

Â example, that I had.

Â I'm spinning about this axis, and I'm slowly precessing around.

Â That's the momentum that I have.

Â So the current angular velocity vector is not equal to just this spin,

Â because you're spinning about b1 and some of the other axis.

Â And there's all these different angles, but

Â you're basically revolving around this while processing around the he vector.

Â So these two motions that I'm doing is basically describe cones.

Â And there's different ways people have written these equations, but

Â you can show mathematically because this is equivalent to,

Â those are the right equations this is a special case.

Â What they call a space cone and a body cone, so

Â this is your momentum factor that's fixed and our whole space.

Â The Omega Vector will tend to evolve around it

Â depending on axis symmetry or not.

Â And then there's the body axis that you have.

Â That one is going to possess around as well.

Â And so, it's like,

Â you can mathematically describe it as one cone rotating on another.

Â And this is a What I want is the largest inertia.

Â So that means it's kind of like a frisbee, a flat plate, spinning but

Â slightly wobbling.

Â That's kind of an oblate condition.

Â If you have a prolate condition, which is more this pen, like a and

Â or rocket body, and is spinning about the axis of symmetry and

Â wobbling, you end up also with a body cone and a space cone.

Â But you can see you're kind of wobbling on the outside of the,

Â the thing's on the outside.

Â Whereas here, it's kind of wobbling around it as a whole.

Â So, if you study some classic papers on torque-free motions, you might come across

Â some of these space cones, body cones and what I hope you remember is wait a minute,

Â I remember this has to do with these intersections.

Â And read up of on other angle rates and how we can rewrite them as conic motions

Â essentially cones and cones rolling this was all good and

Â popular way back before we had all these wonderful computer tools.

Â These days use integrate and [SOUND] out comes the answers.

Â But people done an amazing amount of work in attitude as well to get analytic

Â answers and this is kind of one of the stages where this stuff comes from.

Â