And it says symmetric posi definite matrix that's M is equals to M

transpose it means it's symmetric, greater than 0 means it's posi definite.

Its always going to be true with the mechanical system, not just for

the inertia tensor.

So we can write it this way, so kinetic energy becomes a very convenient

positive definite measure of my errors, which are my rates.

If you go through like Rajon Dynamics or Kaine's equations, or

Dolombert's principle, you end up with these equations that are fairly long

system mass matrix times acceleration.

This would be something like inertia times I omega dot is equal to minus omega tilde,

I omega.

Which is kind of this term plus other stuff and the [INAUDIBLE].

You wouldn't have this in the rigid body dynamics.

Because [INAUDIBLE] is constant.

A simple body frame.

But you get this general description.

This thing is called the Christoffel operator.

If you're very curious it has a lot to do with matrix math.

But it's there, again, the detail is not important.

Q here is just a generalized force.

But you think attitude dynamics, that was the u.

That was the external torque that we had on our dynamics.

If this has translational forces and

rotational torques this would all be in that q vector in a very general way.

So we need a Lyapunov function.

Step one is always, what do you care about?

In this case, I only care about my q dots.

So I want a Lyapunov function that's positive definite only in q dots.

Q happens to appear in here, but it doesn't impact the definiteness of it.

In fact, n, regardless of q, is always symmetric definite.

So, no matter what, like a multi link system.

No matter what I throw in, I'm guaranteed this is a symmetric posi-definite matrix.

And this kinetic energy, essentially what I'm setting it to is equal to my v,

all right?

But over here, so while there's a q here, we don't care about the q, right?

We only care about stability of the rates.

Are the rates coming to zero?

I don’t care if I point left or right.

So, good, we’ve done this, you take a derivative,

this is where I’m going to wave my hands a lot, because there's a lot of math

goes into this, I’m just kind of showing you the pattern, right?

Before we get into the details of [INAUDIBLE] that's where you

have to own it.

That's where you'll be doing it yourself.

This stuff, chain rule, this kind of becomes this with the double dot.

But then M itself depends on Q, which can depend on time and

there's an extra term that you get.

So lots of math.

This takes a few pages putting it all together.

You can write this out, rearrange this.