0:06

Okay now we were looking at an old nominal big omega.zero.

The wheel's not accelerating.

We don't launch that way.

We launch with everything locked down and then we release it.

And in fact, the craft will have some spin after it gets kicked off.

And then we engaged its wheel and momentum happens and

how does it settle in the end as you're spinning up?

There are some classic answers we can go through.

0:30

So To begin with, what's commonly done with these dual spinners,

they're released where it's spinning about an axis that's not aligned with the wheel.

Wheel was aligned with b1, so the spacecraft, let's assume it's spinning

about b2 or b3 or some combination thereof, something orthogonal to b1.

So we want it to be doing this in the end, spinning skinny, but

right now it's just doing this kind of a tumble.

0:55

And then the reaction wheel is spun up.

If this is spinning, and it has inertia of one, and

a spin rate of one, then inertia times spin rate gives momentum of one.

Just non dimensional units.

I'm going to spin up my wheel until its momentum magnitude matches that

of the space craft after launch which is one in this case, right?

Now, in this kind of a dynamical system is Is the angular momentum vector conserved?

1:37

>> The wheel and the spacecraft.

>> Okay so for the system I'm talking about wheel and spacecraft system.

>> Okay. >> Is that an internal torque or

an external torque?

>> It would be an internal.

>> Exactly. So there is still no external torque.

So for the combined hub and wheel system, the total momentum vector which

is what we [LAUGH] used to derive the equations of motion like h dot equal to L.

That h still has to be fixed as seen by inertial observer.

So we can use that.

2:09

So then we want to really study what happens to the attitude as we do this

spin up, right?

And we're going to go through this.

So the initial spin rate, we said, we're tumbling here, right?

About b2, just without loss of generality, I'm tumbling at a rate 1,

inertia 1, gives me momentum 1, that's what I have here.

And spinning up the momentum of the wheel relative to the body,

that was the wheel inertia times big omega.

Big omega was omega of wheel relative to b, right?

2:34

I'm simply having omega dots, that tells me how little h is changing with time.

And we're just going to constantly increase it right now.

We're not doing any input shaping or something like that,

just going to ramp it up linearly until you hit the desired speed.

And the desired speed will make little h be equal to big H, that's what we said.

So the momentum of the wheel relative to the craft is same as the total momentum

of the system.

2:59

So the manuever time you can figure it out.

Well at the end I have to have this much.

That's my constant rate.

You can quickly compute.

This is going to take a hundred seconds, a thousand seconds depending on

how quickly you're ramping up your wheel speed.

Right? This is how long it will take to

reach this total amount of momentum.

So we can compute that.

Nominally we had launching like this, we're spinning right and

then this wheel is going to be spinning up until its momentum magnitude matches

this spacecraft momentum at the time T0, and this is going to line up.

5:03

>> Well no, we are spinning up >> Okay, so you are assuming

>> It's locked in, and we're there, right.

But it is dependent on the direction of these vectors.

Just because little h is equal to the magnitude of the initial big H

doesn't mean the two directions have to be the same.

And that's essentially, this is our initial h that we have, this system.

This is how we launched, wheels were locked down, then we're speeding it up.

5:28

And, at the ends the wheel has a vector

magnitude that is the same as the initial magnitude of the system.

Little h was equal to big H, those are the scalars.

But you could have two vectors that have the same magnitude but

not be in the same direction.

And then this is actually what the spacecraft body must be doing.

So this momentum vector plus this one add up to be the same as the original.

We argue we cannot change initial momentum.

5:57

So, by doing this maneuver, we're not guaranteed, we, typically,

when spacecraft launched this way, they want it to line up like this.

We talked about the Boeing 501s.

They have geostats that always point at the earth, skinny ones like this.

You want them to be nice and even.

They're not doing this big wobble.

They call this coning motion.

This is your coning angle that you can see right here.

So we'd like it to be zero, but it's not typically zero when it happens.

And there's different ways to explain what is happening, but basically this momentum

matching guarantee doesn't enforce that those two things actually line up.

6:33

Now the analysis of this gets rather complicated.

Typically we have to,

now I will show you some numerical results to kind of talk about this.

We don't have nice analytic answers to guarantee these behaviors, but

we can study this stuff.

I mean there's lots of theories behind this but I'm giving it a spin.

Initially with Tom Blake you're not omega 2 but omega 3 for 30 degrees per second.

Have wheel inertia, spacecraft stuff, everything's locked down.

This gives me an initial big H, momentum.

And now I'm going to go ahead and start spinning up that wheel at a constant rate.

So I just took the equations of motion you saw earlier, last class, and

I'm integrating them with these initial conditions and also tracking the attitude.

I'm not just tracking omega but I used Euler angles and, I forget what I used,

something, maybe MRPs.

And at two hundred seconds, I have finished that maneuver.

And you saw then at two hundred seconds, we were close to being lined up like this.

But there was still a little bit of a wobble left, that happens there.

And so

that amount of wobble, that's what's happening with these amplitudes here.

We can predict that now and see what's happening.

So we get close but not quite there.

Here I'm taking a thousand seconds, because I'm taking a lot more time.

So it'll take about a minute to run and

you can see all these simulations running and it keeps wobbling and spinning.

But it's a much more gradual process.

So what's going to happen now in this process is It'll take,

obviously takes more time to roll up, but if you look at the final omega two's

before it was like plus minus seven degrees and even these axis were bigger,

we will have less of a coning angle at the end of this maneuver.

So when It gets close there we'll want to take a look at that.

And that's the general trends we have in these systems.

With a dual spinner we're just spinning it up, we don't want to be too aggressive

because you end up with this huge momentum vector off to the side.

You really have, then you have to do other stuff.

You want to have a coning angle as small as possible so

a little patience goes a long way.

Take a little bit longer to spin this thing up and

at that point, you have a smaller coning angle.

So it will get there.

We're about halfway there.

The way we get rid of this coning angle, you could use thrusters,

in the end if you want to change the attitude.

You could apply h dot equal to l, apply some thrusters, in the right control.

And change that momentum vector of the body, and puff it back into place.

But now we're using fuel, we hate using fuel in space.

There's an elegant method that people have devised which called the mutation damper.

Anybody heard of a mutation damper before?

8:59

No?

What it does is you put basically a ball in a tube of slush,

oil, something highly viscous.

So it creates friction.

So we're losing energy.

All right?

And what we want to have is spin about omega one in the end and so

you make that damper, you line up that tube of slush along either b2 or 3.

So if you're still wobbling with the conic motion,

you're always actually exciting that slosh, and you're losing energy.

And then people have shown this stuff,

at the end it would actually get rid of the coning motion completely.

And that goes there.

We've used this as a 50 turn project in the past where people have to integrate

this and study the spin up and go through that.

It's a fun project actually.

So take away notes is spin up,

just because you match momentum magnitudes doesn't mean you've matched the vectors.

Teaching those vectors and patience goes a long way.

To get rid of the coning angle, you want to put a mutation damper orthogonal.

That'll get rid of the energies of these modes here over time, and

it'll settle in place.

And there's all kinds of, you can look up mutation dampers.

But here's the results.

So with two hundred seconds you see where you are.

A thousand seconds, you know, five times longer,

we've improved it by a factor of two.

There's really, it's an asintotic thing.

So if you want to get even closer, you might take a long time.

That's where, how patient are you, you know.

A little bit of mutation damping can go a long way, and

get rid of the coning motion.

10:23

Other things are, we've seen plots.

I'm just going to go through this, reasonably high-level.

But just to show you, plots have also been applied not just for rigid bodies, but

also systems of rigid bodies like these wheels and dual spinners.

The energy here, this is really the energy of the spacecraft only,

here, not the real energy relative to the spacecraft.

So not the part with i w over two, big omega squared.

So that's called E star.

The total momentum energy of the system is as before, the momentum sphere just

has to be constant, even with internal torques acting because they're internal.

They're not changing the total momentum of the system.

Which is kind of nice, that's a constraint.

So we can rewrite the energy ellipsoid of the spacecraft

in terms of these coordinates.

And with a little math this is how you, this is what you come up with.

So because the energy that you have, the momentum about the one axis is due to

the wheel and due to the spacecraft, so you have to pull out the wheel momentum.

And then divide by the right stuff to come up with the energy terms

of the spacecraft about the one axis.

The two and three axis look like before.

But this is a sphere, this is still an ellipsoid, but it's a shifted ellipsoid.

If your h one two threes are essentially your X, Y, Z, independent coordinates.

If you go look up the classic ellipsoidal equation.

If you didn't have minus h this would be a centered ellipsoid that's there, and

with the minus h, it actually shifts that ellipsoid along the one access.

So as we spin up the wheel that energy ellipsoid is going to move.

And that's what you see here.

Initially we launched the spacecraft spin is here.

We were spinning purely about b3 in that numerical problem I showed you.

Then as we spin up the wheel

you can see these are the momentum coordinates that we have to satisfy.

That's the total momentum coordinates of the spacecraft.

It intersects energy in here.

And the energy ellipsoid starts to shrink inside and of the spacecraft cause as

we spin up the wheel the spacecraft tends to de-spin and change it's orientation.

Although momentum we'd like it to go into the wheels.

So there's different critical points.

At one point this ellypsoid intercepts in this nice figure eight motion.

This kind of a separate trick stuff that we've seen with rigid bodies.

And then when the momentum magnitude matches, this is what you end up with.

Just because you match momentum magnitudes doesn't guarantee that you've despun

the primary body completely.

12:51

And this is where a mutation damper would help get rid of that remaining energy and

make that actually attractive and collapse onto a point.

And that's what would be happening there.

So I'm just kind of showing this more as a pictorial thing.

So you've seen this.

It's not something that'll be on the exam.

But these plots, these graphical visualizations of momentum and

energy are quite powerful.

And if you read a lot of classical attitude control,

you will see these kind of tools being used in different areas.