Casey. >> We have gimbal lock.

>> Gimbal lock, right?

So with the reaction wheels,

the axis about which we produce a torque is fixed as seen by the body.

because its wheel is bolted down, right?

And we had some pictures on this, let me just go down a few.

I like it because it looks beefy, it's really bolted down,

this spin X, the motor axis doesn't change as seen by the body.

So it's always, you put on this 1 Newton meter on this wheel,

it always gives you a torque about that axis, no matter what the body is doing.

Versus if you go look at CMGs, which we had here, now, this is our spin axis.

But then there's a gimbal axis, and that varies with time.

And the gyroscopics we were hearing about earlier,

that's always about that transverse axis.

If this is my spin axis and that's my gimbal axis,

then the change in momentum is about the third axis, that's the GT axis we called.

And that axis varies as you gimbal.

So I could only get that big gyroscopic torque about a single axis.

And what can happen with gimbal lock is with multiple devices,

those axes can all become coplanar.

In fact, that's what would happen if you would just go there,

take the spacecraft, and an astronaut just goes there and keeps pressing against it,

just to be an irritant, that spacecraft will hold an attitude.

But eventually, all the gimbals will line up,

such that all the GT axes are orthogonal to that torque being applied.

And then you have gimbal arc, and everything starts tumbling.

So any momentum exchange devices has some momentum limitations in terms of speeds or

orientations or how you cannot absorb infinite amount of momentum onto a system.

That's just not physically possible.

There's always a finite limit that we're seeing.

So good, this is what we went through.

Now deriving equations of motion,

what was the one overriding equation that drives all this stuff?

>> [INAUDIBLE] >> H dot=L, so Casey,

let's go through this quickly.

H was what for this system?

>> The total angular momentum.

>> Of the dynamical system, right?

And here we have the spacecraft, we have a gimbal structure, and

then we have the wheel within that gimbal structure.

So we actually have three components.

What was the dot, Kevin?

>> The initial derivative.

>> Right, the derivative as seen by the inertial frames.

Immediately transport theorem, right, we're going to have to use that a bunch.

And then the L?

>> External torque.

>> The net external torque acting on it, so motor torques don't go in that L.

Motor torques are internal systems,

where the spacecraft is being pushed off relative to the gimbal frame.

And the gimbal frame pushes off relative to the wheel frame,

those are internal torques.

They don't show up in that L that we had, good.

So, we went through this H dot=L, how hard could that be, right?

And it's a lot of details, a lot of book keeping.

So, some rigor is helpful here.

Just notationally, to review, because we see the same,

[COUGH] excuse me, today, gs is our spin axis of the wheel,

gt is the transverse axis then, and gg, that's the gimbal axis.

So for gimbaling gamma dot, wheel speed, big omega, all right,

that was kind of the rough breakdown.

We had the gimbal frame that we actually used a lot in the derivation.

There's also body frame with a typical b1, 2, and 3, but you see in this system,

it tends to be easier to write everything in the gimbal frame.