0:05

Okay, so let's talk about symmetric stereographic orientation parameters which

contain classical Rodriguez g/ parameters.

The classical had the projection point on the beta not axis but

at the origin and the projection plane was always.

Plus 1 away orthogonal to the beta nod access that you have here and

this MRP's have the projection point on this left outward edge

where it still intersects the unit sphere intersects the beta nod access.

And the projection point is again plus 1 away.

It would be here.

Symmetric sense basically say,

okay instead of making the projection point here.

Or here you can really put it anywhere inside the sphere, all right?

And then the projection plane is always plus one away so

if this moves you always have the other one that's always one plus one away and

it can be a projection plane doesn't have to be inside the sphere.

You could actually be outside that was one question came a plus on so good but

if we do that let's say in this figure you can see where the plane is

we would actually go singular when our courtanians

on the surface are describing an orientation right here right?

That’s where your projection line and your projection plane are parallel,

they never intercept those coordinates go off to infinity.

So, what angle is this roughly?

If this is 90, let’s say this is 30 that would be 120,

times 2, at 240 degree to the right.

I go singular.

The other way is down here and

that’s 240 degrees the other direction, you also go singular.

1:39

It's really 240 degrees about any access.

All it depends is that I get to this point on the beta not space, and

it goes singular.

Beta not doesn't care about which access I rotate.

That's where the symmetry comes in It's like before it was plus minus 180 for

CRPs, plus minus 360 for MRPs.

In between I can make coordinates go really singular at any angle set and

it's asymmetric set plus minus.

2:08

Now let's kind of highlight of this, let's look at the asymmetric sets.

This where things get really little bit weird but asymmetrics said it was,

you know, it's plus minus the same magnitude of an angle and you go singular.

Asymmetric all of a sudden means now it's not going to be plus or

minus the same angle.

It's different stuff, and in fact it's way different.

So here, instead of doing the projection point on the beta non-axis,

we're saying you could really put the projection point anywhere else to.

And so

I'm putting it here on a Beta I axis that could be Beta 1, Beta 2, or Beta 3.

Somewhere along that axis.

I'm doing my projection plane plus one apart and

it's orthogonal to the Beta I axis.

Again there could be first, second, or third Beta axis.

Just not the zeroth one.

And now, we're doing this projection, and we will see what happens.

So, the zero orientation is here, and this would actually project.

If you connect this out,

you can see that zero rotation gives you non zero coordinates.

I made it a little bit weird.

We're more used to simple, if coordinates are zero, then things are coincidental.

That's not the case here.

But more interestingly if you look at this and

go okay I'm making a 30 degree downward, so that's about a 60 degree rotation.

Let say that would be a negative rotation downward,

60 degrees downward and I go singular, okay?

If we go back up 60 degrees upward, I'm perfectly fine.

It seems intercept and give you coordinates, life is good.

3:44

The other point from here is this is same orientation right,instead of 60 degrees

one way it's about 300 degrees the other way,so it's the same orientation but

if you approach it from a different direction is perfectly fine,which is kind

of weird you know before we have always heard hey 60 degrees we go singular.

Doesn't matter if I approach it from left or right.

Here all of sudden if I approach at an orientation,

in a negative sense it goes singular if I approach that same orientation,

in a positive sense, it's fine.

4:15

But then if you continue rotating so this is 360 plus an extra 30 to another 60 so

the 360 plus 60 is 420 degrees, that's a different altitude than the 60.

Then and

you approach it from that direction, you know now you're going to go singular.

The entry point which is the same orientation just as we approached it

we didn't do a revolution to get there is perfectly fine.

4:40

So hope that gives you a headache, it gave me a headache when I started looking into

this stuff, wait a minute it all depends right?

But there's some unique stuff we can do with this.

because all of a sudden, it's not here with beta not.

It doesn't care about which axis you're rotating about.

With beta 1, if you look at the beta 1 coordinates, b1,

beta 1 is e1 times sine phi over 2.

It depends on how much you're rotating about e1.

If you're rotating purely about the 2 axis or the 3 axis, and

e1 is always 0, and you never actually reach this points.

5:13

So, with asymmetric sets you can

get rather off behaviours as far as where this singularities occur.

But also it matters about which access you rotate.

Yes?

>> What would be the application and the coordinates like these?

5:47

So that means this point was actually being moved all the way on the extremum.

That means I'm only going to go singular if my attitudes go here or

here and at this intersection point,

beta one being minus one, means all the other betas have to be zero.

Otherwise, we're not a unit constraint surface, alright.

6:20

But it has to be a pure rotation about your B1 axis because beta 1 is 1.

It's only rotating about that 1 axis.

And then you can write up your coordinates.

I'm calling them eta.

In this case, this is nice because with the three parameters set now

I can actually describe a generally tumbling body.

6:40

Let's say this is my one axis.

The symmetry axis of this one, and two and three come of orthogonal of this.

If noticing this can be normally tumbling like this away from that one axis.

I can used a free parameter set that could never go singular

6:55

unless it changes its behaviour and start spinning here whereas with

the at some angles enough I'll go singular and

we can switch which is nice, but it gives you a discontinuous description.

With these I can now describe the motion of this with a three parameter set.

In a completely continuous way I never go singular unless I start change to tumble

and start to spin perfectly about that B1 axis in which case yes,

you will hit the singularity point.

So that's a particular application how you could design these things now.

7:30

So you can go through to math this isn't a book, we're not covering in much but

there is mapping to to betas and there's actually whole papers on how to do this

differently, but it goes singular at two different sets of angles, this is no

isometric, and there's also a shorter set which means we can switch between them.

And so here's a time history, I'm doing something that is tumbling, it's tumbling

quite a bit and you can see here the [INAUDIBLE] time history looks fine.

The more interesting is this plot actually, where we're showing,

this is the same tumbling motion so it goes past 180, past 360, and so forth.

It's just not tumbling about this one axis that we've chosen to put our

projection point on.

8:11

And the segments go singular.

You can see them going around.

They all complete this is kind of a polar plopped.

So the angle here you can see your principle rotation angle and

the magnitude is the magnitude of this cordinance.

So the MRP's go to infinity as you completed a 360 spin.

8:28

The Q's the CRP's go singular as you get to the 180 point and

those coordinates go off to infinity.

Whereas the adus,

we're doing the same tumbling motions, you get a little lot behaviors but

it's doing stuff and staying bounded and continuous all the way through.

So that's kind of an illustration of that.

9:13

That means if I give you a set of symmetric stereographic orientation

parameters is there only one set of attitudes that it responds to,

something we have to look at.

So this is the attitude here, right?

You project it, here's your projection point.

This is my set of SSOPs.

9:41

But this projection line between this set of coordinates,

this axis, if you keep on going there's a point out here If my beta's are here and

I make a line with this projection point,

I get the same line that will intercept to the same point on that plane.

10:03

So, student's who use these, you have to be careful.

There are some papers written on this, on how to use these coordinates to do

a constrained altitude control, never exceeds 240 degrees,

10:16

but if you do let it go past 240 you're someone in trouble because then all of

a sudden we cannot differentiate between, there's two possible orientation which

respond to, one is past 240 the singularity, one is less than that.

And we're always sticking with the ones that's less than that.

That's kind of the applications of these things.

So, as you get into this coordinate design,

quickly take a look at these issues.

If you move a outside of the sphere, you immediately have ambiguities always,

which is a bad thing.

You'll never quite know this could be this orientation or

a completely different orientation.

And so that's why we don't put projection points outside of that.

10:52

There were other papers that I actually came out of teaching this class

that were done.

Here's one by Jeff Marlin that we did, Instead of putting the projection point

and you were here, we're moving it along the surface.

And always being orthogonal centered in the origin.

And that gave you whole family of parameters that kind of encompasses

the MRP behavior but also those asymmetric stereographic orientation parameters.

And there's a really beautiful math that goes in this that gives you.

The equation ends up being essentially the same form regardless of where you put it.

It's just this a parameter comes in and so

that you can write it on the very general way.

And so anyway also published the paper recently

about taking the surface and looking at different geographic

mapping technics and how do you and how do all these things relate and

they all have benefits and drawbacks and you know coverage and.

Different things that you can do with it, so

anyway there is a whole field out there people continue to publish.

11:54

Yes sir?

>> So you kind of mentioned optimizing

our set of singular so that when try to hit that if we know the rotation.

How practical is it to change where that singular point is in real time,

as your rotation changes, so that you kind of constantly adapt.

>> Yeah. I haven't used them too much,

to be honest.

My primarily coordinates of use are MRPs.

I find that I can do anything.

I just have to deal with the discontinuity.

Here, if you're doing this, you have to make sure you using them in situation

where you know you're tumbling and this is the nominal tumble and

you just not going to tumble about this other axis,

it's going to be a different kind of tumble.

If it does switch you're going to have to switch to different coordinates that don't

go singular if you had to do that.

From a feedback control perspective this is interesting because again we pointed

out that if you have coordinates that go singular at 90 degrees and

you definitely want to make sure if a solar panels never go pass 90 degrees to

the sun you get this increased stiffness and we can still analytically prove

stability of those controls which is kind of a nice thing.

But then if you deal with finite actuation then it's harder to guarantee that

somebody didn't kick you off hard enough off the rocket and

that you're still going to tumble past it.

And so you have to make sure that you can handle that.

13:10

Life is full of details.

But you should be able to understand them at least.

Yes, Jordan.

>> Has anyone ever looked at using a projection surface over than a plane,

maybe something with some curvature to it.

Could see maybe how you could- >> You're just determined

to make my life difficult, aren't you.

>> [INAUDIBLE] >> Anybody ever looked at that?

Ever is a big word.

>> Yeah.

13:38

>> You think you could avoid singularities by getting an object with some

curvature to it.

>> You might, especially with the MRP like thing, I mean could do something else.

You might have a different surface that then gives

you an even more linear behaviour.

I think that's the biggest benefit I would see of that is to make these projected

coordinates even more linear, maybe, I mean does the Higher Order Rodriguez but

I haven't quite seen such a stereographic representation of that,

maybe that's something that they're doing there implicitly.

So, no, that could be a whole paper right there, we'll see.

Lots of fun little nuggets.

Nothing else, you go to these conferences, you blow everybody's minds,

because this is something people don't typically focus on.

They just go for quaternions and then don't even know what quaternions are.

If you tell them there's two sets, they're completely confused.

And Hopefully you'll have a much better knowledge of this.

But there's an interesting world of kinematics and

how we can describe orientations.

So yeah, nice idea.

We should maybe explore that.