If you look at complicated systems, it just gets even more complicated.

You add an extra thing, it gets even more complicated.

In astrodynamics, in translation and in attitude.

It's amazing how often with the right formulation, the right kinematics,

we end up incredibly elegant,

simple solutions that comes out of this stuff which, to me, is very exciting.

So occasionally we get a little carrot,

it's not just being beat over the head, prove all the parameters from property.

But at the end, it was a beautiful simple one.

This was another case, so I want to show you, outline you, how this math and

logic works.

You won't be following all the details of the substeps.

But the logic of how we have to implement it, and why these quaternion properties,

and also vice versa similar MRP properties can be exploited,

becomes very apparent in this formulation.

So, that's why I've chosen to kind of end the control discussion with this one.

So our desired thing here is I'm going to follow up.

Which did it in terms of quaternions.

So their epsilon is beta one, two, and three,

the vectorial part of the quaternion.

And they want a perfectly linear closed-loop dynamics.

And we have this dynamical system.

So the question is what control u do we have to create

such that the error response perfectly satisfies this?

And this becomes basically an inverse dynamics approach.

We start out with a desired closed-loop dynamics and

then we have to back substitute everything and solve all the algebra back for u.

So right now here there's no u in site so we have to get there.

All right?

So this is a step.

How do we make this work?

Well this is the definition of epsilon, you can see the classic definition.

If you go back and look in our quaternion notes, there was some other formulations,

this is the T matrix, it's a three by three.

The quaternion, the beta dots had these four by three times omega so

that's that lower three by three part.

Basically that's where that T matrix is.

It's a three by three matrix defined here and you can see it's a diagonal of betas

so zero times a diagonal of identity and the other three that's actually nothing

but a skew-symmetric matrix, that's nothing but epsilon tilde.

So, different forms.

You can look it up.

Nothing too fancy yet.