So now lets look at what this actually means, this mapping.

We have Beta I over Beta naught CK was done by the hyper surface, all right.

That's my artistic impression of a full dimensional hyper surface.

I'm just drawing a circle.

In essence, what I'm doing is doing a one D rotation.

If you only rotate about b1 for example, then only the Beta 0 and

the Beta 1 parameters will be non zero.

The other two will just be all zero.

So you can think of this at interpretation just doing a one dimensional rotation

then, your four dimensional hyper surface just boils down to a circle and

circles are so much easier to draw in 2D than 4D surfaces.

But it really, I'm trying Beta I, everyone of these axis are actually orthogonal and

that four dimensional space so once you've figured out two of them in a plane

the same math holds across the other planes as well the geometry.

So this is interesting.

Actually for the MRPs to [INAUDIBLE] credit he is now professor

at Georgia tech but he and I co-wrote some papers when he was on the post doc and

I was just finishing up my PhD.

And in one of his papers,

he actually interpreted the MRP's as being a stereographic projection and

he put a certain point, and that's what this math means.

I thought, that's kind of interesting.

But what does that really mean?

And so I started drawing it out.

So, there is a whole family of parameters that are developed way back then,

that people have been following up on now.

And they're all based on the same principle.

So as CK was saying, your quaternion has to resolve itself to be on the surface.

You don't just have the attitude that all of a sudden, leaves the surface,

goes inside, and wanders around.

You have to be on that surface so when we integrate,

we always have to make sure to map position.

We stay on that surface.

Good, but this is a four dimensional manifold.

If you want to reduce this down to a three dimensional amount of space,

we have to do a mapping.

And so for cartography we use the Earth, all right?

And you could take different maps but we take the Earth's surface and

you essentially peel off that surface and we stretch it onto a plane.

And every one of those maps has issues, right?

The classic map has issues where?

>> Poles.

>> The poles, those areas look huge, right?

Greenland looks amazingly big and then you see it actually on a globe,

and that's still big but not quite that big, right?

Things get stretched way outta proportion.

There are more area preserving projections that happen but somewhere there's always

a single layer, there were things go a little bit wonky, right?

So there's different mathematical procedures on how did map

the earth surface into a plane.

This is one of them and it's called the Stereographic Projection.

[INAUDIBLE] recently published a paper looking at whole series

of geographic projection methods and how this map, it applies itself to coordinates

as well kind of based on this work which is kind of cool, very, very elegant.

So what we do here?

With a stereographic projection you have to have essentially a projection point.

Think of it as a point light source.

You've got your point here,

that's my point on the surface that represents, let's say this is 45 degrees,

this is always angles over two, that's how we define datas.

So that would be like a 90 degree rotation, that's one point on that

surface, I want to project that point, I make a line between that

projection point and the attitude of interest, that's my projection line.

And let it intercept a projection plane.

This plane is define by Beta Naught equal to plus one.

So that actually that means Beta one, Beta two, Beta three can be anything.

It's a plane orthogonal to your first axis, the Beta Naught axis.

That's it.

So this is one D rotation that plane just becomes a line.

Instead of a three dimensional hyper space, but

you could imagine two other dimensions sticking out of here.

Just very hard to draw.

So if you do this, you can show, in the homework actually you do,

you can use geometry and symmetry triangles, it's not very hard actually.

Once you see the pattern, that this mathematics is true.

This projection, this coordinate, in this case, if I have Beta 1,

I would get q 1 here.

I would get my classical Rodrigues parameters.

So you can think of the CRP as the hyper surface stereographic projection

onto this hyper plane that happens to be located here, okay.

And there's different ways we can play with this.

Let me go, let's try to get these things up and running.

There we go.

So hopefully, yep, this works.

So on my website, if you go to classes 5010,

you can get to these links they're also in the slides.

I've got a little JavaScript exported a while back.

So I've got two points I'm tracking here.

The red point is my current attitude and then this point up here.

This is the projected attitude onto that hyper plane.

That's the CRP coordinate.

So what happens now is, I can make rotations and you can see for

small rotations, as I double my angle, I roughly double my distance.

That's good.

I double my angle, I still double my distance.

But as you get further away, if I double my angle from here to here,

I haven't exactly doubled my distance, right?

That one goes off to infinity.

This angle is the singularity point that you would have.

I can't move it.

Darn, it will be nice.

So 90 times 2, this are all half angle.

That's the upside down condition that you were identifying earlier to be

singularity.

So, geometrically, that projection line basically became parallel with your

projection hyperplane and they will never intercept.

And that's the cue.

They went off to infinity in that direction.

That's what happened there.

Good.

Any questions in this geometry?

When you see this a few times and you'll play with it in the homework and you see

some other coordinates arrived on this, there's some interesting stuff you can do.

So what we like is for smaller angles, and it's not just ten,

20 degrees like with older angles, you can do extra 30 large angles.

They behave quite nearly, but if you go close past 90 to 180,

very non-linear, Kayleigh.

>> [INAUDIBLE] >> Yes.

>> [INAUDIBLE] >> Okay.

>> [INAUDIBLE] >> I'm getting in to that,

I havent talked about that yet.

>> Okay.

>> I'm glad you're paying attention because we're about to get to that.

What is that q S, right?

So I'm just talking about one set right now.

So lets also remember, right, quatronions are not unique.

There's always two sets of quatronions.

There's the Beta set and the minus Beta, so

whatever point on that sphere, the opposite is equally valid and

sometimes preferred depending on how far you've tumbled.

One is long, one is short.

And in this case,

everything that's the short location in quaternions is the stuff on this side of

this sphere because I'm always drawing the Beta Naught axis on the horizontal, right.

So this is always the short rotation and

then when the other expand out of this plane.

So we have one set of quaternions that we're mapping onto a CRP.

Now, if I use the other set of quaternions,

it's in the entry point, that's way down here.

If I do the same projection, you can see because of this two points on

opposite locations and the projection point is in the middle of this sphere,

it doesn't matter if I project Beta or minus Beta.

They all project on to the same point.

So these ultimate set for Rodrigues parameters, I call the Shadow Sets.

Here the shadow set doesn't really do anything, they are the same.

So thus the CRPs are unique.

Quaternions were not unique, that is two possible sets.

But the CRP with just one free parameter set and its a unique set.

There's no way to avoid the singularity.

That will be different with the MRPs that we'll talk about shortly.

So that's where the s comes from.

So in this case you can see geometrically, yep, they always will project.

And if I go to 180, it goes to infinity.

If I go to infinity and beyond as, who quoted that?

>> [LAUGH] Lightyear.

>> Thank you.

Good. Buzz Lightyear.

Infinity and beyond that's a wonderful statement.

Definitely works here.

Cause the coordinates literally do good to infinity and beyond and

then it come back in the minus sign and the form of the infinity and continue.

It's just we can't handle infinity very well in computer.

Well that's just the challenge.

So good!

Moving on, so we've got that and that's a shadow set.

If you go back to the definition of them, your Betas, this is very apparent.

Since it's Beta I over Beta Naught,

the alternate quaternion set is minus Beta I over minus Beta Naught, and

those minus signs cancel, so that mapping gives you the same set.

So CRPs unique at singular at 180.

Okay, so we just discussed that for the visualization.

And we did this and that's, okay there we were.

So we've got this mapping here.

So the singularities cannot be avoided with this three parameter set.

10: CRP Stereographic Projection

Kinematics: Describing the Motions of Spacecraft

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