So what is the full mapping?

We can always go to the DCM.

You guys are now experts.

We can look up.

You can derive them, or you can look right at the appendix in my book.

You can find all of the sets.

Somebody's giving you a 3-2-1s, and somebody else is giving you 1-2-1.

And you just map all of them into the DCM form, with the right.

And this goes into this frame, into this frame, this frame to this frame.

And then you carry out the multiplication and after you have the answer,

this is the b relative to n and somebody wants then a third frame, a third set of

order angles, they want three one two, something we haven't talked about, great.

Then you have to find the right formulas to extract in this process how to go from

the DCM to the right order angles.

So the DCM you can see is really at the heart of many of these attitude

descriptions.

It allows me to translate, as we were talking earlier, if somebody has a t3-2-1,

a 1-2-3-1-3, I would just take the 3-2-1, map it to a DCM,

and then from the DCM, I have formulas to map it back, all right?

That always works.

Sometimes there's faster ways, but that's always possible.

And for addition-subtractions, mapping 321s and 321s,

all the DCMs do the multiplication and then map extract will always work.

And it is good to numerical precision.

You're not making any approximations here.

So that's why in code, we always want to do the full thing.

Don't do the linear stuff.

But sometimes in analysis we're trying to predict well just how

is that closed group response going to be once we get close to the attitude.

Is it going to keep oscillating forever and take a long time to settle?

That's where the linearization might be handy, and

we'll deal with some approximations of them.

But this is the full answer that we have.

So we can mix and go back and forth, as needed.

If you have mixed coordinates, [SOUND] I

don't think I've even actually seen nice, elegant, compact formulas.

People typically just mix.

If they have something in quaternion, some of you have heard of quaternions, some

of them in Euler angles and have to add them, I'm just mapping them all to DCMs.

because then I can deal in DCMs and then I extract out what I need.

So now we know how to do addition.

We know how to do subtraction the same way.

We do it with DCMs.

It's just the key is the mapping to and from of areas.

So subtraction have this the same kind of a process you take your angles you

have b relative to n, r relative to n, and I'm looking for b relative to r.

I have to just map these all into the equivalent,

DCM formulas then I do my br is bn times rn transposed because rn transposed is nr.

So bn nr gives me br, right?

So that two letter stuff,

as long as you have good bookkeeping, no problem, you will get there.