Now, let's talk about attitude. With the axis symmetry I got some really easy looking omegas. One of the most flatline is gets you the sine and cosines. [SOUND]. That's how we get the amplitude and the right phase. But that's pretty straight forward, right? So, but the resulting motion, just looking at these yaw, pitch, rolls relative to inertial frame, they were all over the place, and very nasty to predict. Lots of wiggles, bumps, and stuff that was going on. So, it wasn't very intuitive to predict what happens to the natural procession of a torque-free attitude motion. So this is a classic result that people have done. And anybody heard of mutation procession before? Good? Was it Russell? What does mutation mean? >> It's where you're like slowly going around and circling around what you're [CROSSTALK] >> So is this mutating or is this processing? Is it mutating now or processing? Who thinks mutation procession definitions are confusing? >> Like slowly around the depth. >> Good. I hate them, quite frankly. I'm always confused by those darn things. Why? In the end, those mutation procession things, as I'll show you, they're nothing but older angle rates. And, as often in the literature, we're not consistent. Some people talk about 3-1-3 Euler angle rates which means certain motions mean something. That's the notation part. That's the procession part. It's a 3-1-3 versus a 3-2-1. The processions are often very similar because they are both 3 first axis. But the 3-2-1s and the 3-1-3s are the two common things you will see people use to describe torque free attitude motions and look at the tumbles. But they don't always tell you what they are doing until you look at the math, it takes a long time so I'd rather they just tell me, look I'm studying it using three one three over under rates. I know right away what you are talking about verses mutation and damping. But it is a common mutation, common thing you hear so let's cover it and I want to show you a little bit what it does. The other issue is so we are going to use older angle rates. to start to prescribe how much does this wobbling disk process around and what's happening there. The other trick is, whereas my earlier example acts as a metric, I was plotting my yaw pitch roll angles relative to the inertial frame. With nutation precession, those terms, we're never talking about attitude motion relative to inertial. We're talking about attitude motion relative to your momentum vector. That's the key difference. So, if this thing is spinning this way, you have a big momentum vector sticking up. Great! Then you talk about precessing about that momentum vector and how much you're nutating and wobbling Around the momentum. If you're spinning this way, then procession is really defined around that momentum vector, right? So it has nothing to do with where your inertial system is, it's torque-free motion. Momentum vector is preserved. So, if we do that, let's go through this. I know some of you have seen this. So, you have some momentum vector, it's torque-free motion, as seen by inertial observer, this is a fixed quantity. You give something some spin It's going to have some fixed momentum. And whatever else happens and how it gyrates and wobbles, that momentum has to be fixed. So, we can now pick an inertial frame that is lined up with n3. Just historically they pick the third one for some reason, to be = -H. So now, I can talk about attitude motion relative to this frame here. And n one two are really arbitrary, they don't really matter much in the end you will find. So I can write the momentum vector as minus h times n three and this is also an frame, right? It's just lined, however you released it, that momentum was this direction. So n three goes in the opposite, that's it. So now we can write that and we can say okay, good this makes my h vector in this particular end frame, all right this is a special end frame, not a general end frame, is zero zero something times a dcm. This allows me to write my momentum vector components in the body frame in an easier way. Generally this H if you have an arbitrary inertial frame H would have an H1 two and three in frame component. Here we only have a third component, so that means times the DCM you are not only pulling out the parts of the third column. That saves you lots of Algebra. Why do we do that? So here's the DCM and using 3-2-1 Euler angles here so I'll get 3-2-1 Euler angle rates out of this stuff times that zero zero minus H. You do this, you have the H times sine and the minus minus cancel. Here we have H with a minus and a sine and a cosine of the third angle, which is rolled and the second angle which is pitch. So phi is rolled, theta is pitch, and the same thing for the third term. And this is the momentum vector in the body frame. But then, you remember from kinetics, H is also equal to i omega, right? That's what we derived for rotational motion. And in principle coordinate frame. This is simply i one, omega one, i two, omega two, i three, omega three. So this is an elegant way that we can use the three dimensional and get momentum vector to provide three constraints on how does omega and the angles have to roll it. And you can see out of the 3-2-1 Euler angles pulling out this column here because it was all about the third one, theta one doesn't appear here. There's no yaw angle in here. Which kind of makes sense because there will be the motion about the momentum ,vector that happens there. So, if this is the spinning and the processing rate you're going to have, if we take the system and twist it for 80 degrees about the momentum vector, it's completely asymmetric. It gives you the same kind of a wobbly motion. So that's kind of what manifests in the Mathew, does know we are. There is H and these omegas. Well, this is nice, so I don't have to integrate differential kinematic equations to come up with, at this attitude, what is omega going to be? It's kind of like the energy conservation, the bouncing ball problem, right? That we've used the complete three dimensional h vector here instead of just the magnitude as we did with pole plots. So that's good. So now, we can divide by inertia and you get this expression. So, I can write analytically, if this is the pitch and this is the roll, these are the omegas you must have or otherwise you're violating momentum. So good, we have that. Now the next step is we have our differential kinematic equations for my Euler angle rates and omega. Now I just had an expression of omegas in terms of these angles and you plug that in so instead of having a differential kinematic for theta, you know, your state rates then you differentiate omega to get the accelerations, here we used angular momentum to completely get rid of omega dots and I get, I end up with torque free motion that's three coupled non linear. But first order of differential equations. So I got, I used angular momentum to get rid of some of those dots. Didn't have double dots in the end, in attitude angles, I only have single dots. And this would be your procession, this would be your mutation if you used a 3-2-1. And then, the third roll, we often just call roll as the final roll motion. But that's the wobbly plate problem, we've all seen this plate on a table, right, that wobbles It doesn't keep the same plane, that plate kind of goes around and precession and does some weird stuff, right? That's all going to be predicted now by these motions. If you look at this, positive, positive, H magnitude is positive, sin squared, cosine squared, all positive. The precession rate, your yaw rate essentially, is always negative. And this comes out of the definition that we've chosen and end frame which is in the minus H direction otherwise there will be a sign flip. So is different this could be different but this is more the classic one whereas the pitching, that's basically a mutation rate. Does this, if this spinning, is there an up and down motion that happens because of this? And you can see, for example, if i3 is equal to i2. This term actually vanishes and you get rid of notation rates, as well. So different shapes call certain rate things to change. Anyway, so this is the classic one for general inertias, and we cannot be positive. If you have an axis symmetric case, so I'm just assuming here B1 is my axis of symmetry. So, I1 is unique and I2 and I3 are equal. And you plug it into those equations. They simply fly down to a constant mutation rate and a constant procession rate that you would have. This would tell you that if the sphere is spinning, that at about the momentum vector, this object is spinning, but it's also slowly processing around. It has to do that to conserve angular momentum. And so, these are some classic results where you can now actually make predictions about the attitude motions, and turns out all their angles are kind of convenient for some of this stuff. They're nicely to be worked out. And this tells you what happens to the motion, since data is constant, your roll rate that you would have in this situation also will stay constant. So this is pretty classical. We build a lot of spaceships that are kind of, at least originally, almost cylindrical, very axi-symmetric just because they come out of rockets and it's been released and have a wobble. These equations are very common to kind of analyze what happens post tip off, how are things going to process, what's going to go on. So, it's a convenient form. Now, there's some other stuff people. I'm just going to highlight this. We're not going to go through details. There's no particular homeworks on this. But if you have angular momentum vector, here, this is my axis of symmetry, for example, that I had. I'm spinning about this axis, and I'm slowly precessing around. That's the momentum that I have. So the current angular velocity vector is not equal to just this spin, because you're spinning about b1 and some of the other axis. And there's all these different angles, but you're basically revolving around this while processing around the he vector. So these two motions that I'm doing is basically describe cones. And there's different ways people have written these equations, but you can show mathematically because this is equivalent to, those are the right equations this is a special case. What they call a space cone and a body cone, so this is your momentum factor that's fixed and our whole space. The Omega Vector will tend to evolve around it depending on axis symmetry or not. And then there's the body axis that you have. That one is going to possess around as well. And so, it's like, you can mathematically describe it as one cone rotating on another. And this is a What I want is the largest inertia. So that means it's kind of like a frisbee, a flat plate, spinning but slightly wobbling. That's kind of an oblate condition. If you have a prolate condition, which is more this pen, like a and or rocket body, and is spinning about the axis of symmetry and wobbling, you end up also with a body cone and a space cone. But you can see you're kind of wobbling on the outside of the, the thing's on the outside. Whereas here, it's kind of wobbling around it as a whole. So, if you study some classic papers on torque-free motions, you might come across some of these space cones, body cones and what I hope you remember is wait a minute, I remember this has to do with these intersections. And read up of on other angle rates and how we can rewrite them as conic motions essentially cones and cones rolling this was all good and popular way back before we had all these wonderful computer tools. These days use integrate and [SOUND] out comes the answers. But people done an amazing amount of work in attitude as well to get analytic answers and this is kind of one of the stages where this stuff comes from.
