My attitude area is about 90 degrees.

If I'm really tired, and I'm talking like this all lecture long,

you should probably throw tomatoes at me.

But my attitude there is 180 degrees.

If I keep on wandering and start talking this way,

my attitude there is 270 degrees.

Is 270 worse than 180?

Sorry, what was your name?

>> Rick.

>> Rick.

Brett, sorry.

>> Brett, sorry.

>> What do you mean by worse?

>> I'm making the numbers bigger.

This was zero, this was good.

90, 180, 270, it sounds like 270's worse than 180.

>> Better.

>> It's actually better, right.

So the fundamental thing that's built into attitudes is there is a finite set.

Mathematically, it's called the SO3 groups.

If you read papers in attitude stuff and see SO with the 3,

it's a mathematical description of this group.

That's what we belong to, not the attitude coordinates.

They're not part of a vector group, where you can do vector addition,

subtractions and so forth.

SO3, so there's a finite limit.

You can rotate, rotate, make it bad, get as bad as it gets, and that's 180 degrees.

You would never have infinite angle, because at some point,

you're going to revolve again about the orientation you want to have.

This is really important in control.

You will see this when we try to do detone in control.

Some descriptions automatically take advantage of this, others not so

much, right?

And so depending on what you're doing, they have different benefits and

drawbacks.

This is very practical, so I can tell you, teaching skydiving, right?

Or hanging out, outside the plane.

It's a high winged plane, we had a Cessna 182, little strut.

The student stands out there, this is an AFF program, solid free fall.

One jump master holding on the left, one jump master holding on the right, and

then we are supposed to leave together.

The student's supposed to up, down, and step gently off into the wind arch.

Put hips forward, arms back, legs back, so

you kind of fly like a butterfly, t that's a theory.

The reality, lot of profanity, [NOISE] pushing, shoving, kicking and off you go.

And there's at this one video I got in California that was really cool.

I did this jump at the jump master, we went and

the student really shoved off hard.

And we're both trying to stabilize it, but there was so

much momentum put into the system what was basically going up side down and

look over smiling at each other, going what the hell?

And we just basically complete the tumble, flatten out, fly on the rest,

student had no clue.

This was their first jump, you get blackouts and everything.

But that's basically it, instead of trying to unwind it and fight it all the way,

at some point you're going kind of attitude perspective.

It's easier to just hlip it all the way around and recover here again,

if you're unconstrained.

If you're a mechanical system that has wires attached and you can only rotate so

far before you rip out wires, well then it really matters.

You really want to make sure you don't just cut that thing loose.

But in space, we're typically tumbling freely, there's no constraints.

So again, these tumbling things,

you will see something throughout the class we keep talking about.

Why did we describes one way versus another?

Which one helps us into control, which helps us in the dynamic prediction?

Attitude errors can only get up to 180 degrees.

4 Truths, need a minimum of three coordinates.

We live in a three dimensional world, right?

So your position has three coordinates thing, but

also your attitude has three degrees of freedom.

So your orientation needs a minimum of three coordinates to

define any general attitude, okay?

That's easy, but the minimum sets of three coordinates.

We'll have at least one geometric

orientation where the description becomes singular.

And as we just mentioned, singular might mean ambiguities,

because zero over zero in the math.

Because this angle could be an infinity of angles.

And you will see different cases of that, or

it means your coordinates literally go off to infinity.

You've got something finite over zero at that orientation, and now big,

big numbers.

So both ways you have issues, those are called singularities.

So at or near the singularity,

the kinematic differential equations are also singular.

We're going to develop this equation, this is basically the equation that relates

this omega that we've developed, the angular velocity vector omega.

That's what we measured in [INAUDIBLE].

That's what we use to actually develop and

derive our equations of motion in the future.

But when we integrate and have yaw angle,

pitch angle or quaternion rates or DCM rates.

We have attitude coordinate rates that has to relate to these omega,

that's our differential kinematic equation.

We can see it over and over and over and over again.

But that's the equation that would go zero over zero or

something over zero which makes this whole thing blow up.

Now this singularities or these zero issues can be

avoided by moving to more than three coordinates.

So you need three just to do generally tumbling objects and there's coordinate

sets that can avoid singularities at the cost of additional coordinates.

Anybody here heard of coordinates that's non-singular?

>> Quaternions.

>> Quaternions, Eulers parameters is one of them.

And there's different sets we'll see, actually.

There's several sets that come up here that'll be non-singular,

but it's always four or more.

So let me invert that, several of you have heard this before.

If I have a set with four coordinates, am I guaranteed, it's non-singular?

[COUGH] Luis?

>> Yeah.

>> No, because you could just have a redundant set that doesn't-

>> Yeah.

>> Add any.

>> So just having four coordinates doesn't guarantee that you're non-singular.

But it allows the option of having a smart set and there's lots of different sets.

And chapter three, we'll be going through most of them,

not quite all of them because there's a bunch of them that do that.

So we have to always go to more than four.