So we don't want to apply this to a problem if its tumbling and

you have to account for

a multi revolution problem this would be a poor choice of coordinates.

But compared to the Euler angles, Charles,

with Euler angles how far away were we from the singularity at the best case.

>> Is it 90?

>> 90 degrees.

You can pitch up and down 90 degrees.

So the best is zero pitch.

Or inclinations, you go from zero to 180.

So a polar orbit, 90 degree inclination,

is the furthest away from those singular behaviours.

That's it.

Here at least we have a three parameter set that moves a singularity 180

degrees away.

So you could see it right away using these if you lean your eyes dome,

their domain of linalization where it's valid it's much bigger about this twice

the motor oiler angles are doing.

So we're going to speed up to looking at how lineariable these coordinates are.

The relationship back to if you plug in the definition this EI times sine for

two this is cosine over two, so

sine over cosine just gives you tangent X this vectorial part to E1, E2 and E3.

So, that's a nice vector matrix notation that puts it together.

And if

you linearize

CRPs.

If you're just doing little wiggles and tracking errors and stuff.

And q1, 2's, and 3's, you take the values times two, that's roughly how many radians

error you have, in roll, pitch, yaw, that your kind of more familiar with.

So there are ways to interpret it, and

again if the magnitude gets large I know by the way this is close to 180.

That point so,

there's direct ways here that we can map easily between this coordinates.

You can always go to and in from the DCM.

But these are much faster direct relations, especially for analysis.

And again, in this function, if you have 180 over 2 is tangent of 90.

And tangent 90, that's where that curve goes off to infinity, right?

That was the issues that we have.

Good, so much better suited for

large rotations compared to all other angles, but is still doesn't quite allow

a completely tumbling body without hitting a singularity, right?

That's still the remaining issue with this, but definitely an improvement.

Now, from the from a DCM, here's some convenient formulas.

I show you another one in a second.

But if you have to program it up,

there's different ways to compute this particular term in terms of betas.

And you take C transpose- C divided by that squared.

You can get this tilde, or

here's another way to get it, from the DCM you can get the q1, 2s, and 3s.

They're all kind of piggy backing on,

this looks similar to how we got order parameters from a DCM.

There was difference in the of diagonal terms to get the vectorial part but

the scaling looks different.

So these things are all related.

But those are some convenient form as you might see if you're using it.