0:05

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

Â contain classical Rodriguez g/ parameters.

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

Â at the origin and the projection plane was always.

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

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

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

Â And the projection point is again plus 1 away.

Â It would be here.

Â Symmetric sense basically say,

Â okay instead of making the projection point here.

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

Â And then the projection plane is always plus one away so

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

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

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

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

Â we would actually go singular when our courtanians

Â on the surface are describing an orientation right here right?

Â Thatâ€™s where your projection line and your projection plane are parallel,

Â they never intercept those coordinates go off to infinity.

Â So, what angle is this roughly?

Â If this is 90, letâ€™s say this is 30 that would be 120,

Â times 2, at 240 degree to the right.

Â I go singular.

Â The other way is down here and

Â thatâ€™s 240 degrees the other direction, you also go singular.

Â 1:39

It's really 240 degrees about any access.

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

Â it goes singular.

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

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

Â CRPs, plus minus 360 for MRPs.

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

Â it's asymmetric set plus minus.

Â 2:08

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

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

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

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

Â minus the same angle.

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

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

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

Â And so

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

Â Somewhere along that axis.

Â I'm doing my projection plane plus one apart and

Â it's orthogonal to the Beta I axis.

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

Â Just not the zeroth one.

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

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

Â If you connect this out,

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

Â I made it a little bit weird.

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

Â That's not the case here.

Â But more interestingly if you look at this and

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

Â Let say that would be a negative rotation downward,

Â 60 degrees downward and I go singular, okay?

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

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

Â 3:44

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

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

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

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

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

Â Here all of sudden if I approach at an orientation,

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

Â in a positive sense, it's fine.

Â 4:15

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

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

Â Then and

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

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

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

Â 4:40

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

Â this stuff, wait a minute it all depends right?

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

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

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

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

Â beta 1 is e1 times sine phi over 2.

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

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

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

Â 5:13

So, with asymmetric sets you can

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

Â But also it matters about which access you rotate.

Â Yes?

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

Â 5:47

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

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

Â here and at this intersection point,

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

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

Â 6:20

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

Â It's only rotating about that 1 axis.

Â And then you can write up your coordinates.

Â I'm calling them eta.

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

Â I can actually describe a generally tumbling body.

Â 6:40

Let's say this is my one axis.

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

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

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

Â 6:55

unless it changes its behaviour and start spinning here whereas with

Â the at some angles enough I'll go singular and

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

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

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

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

Â you will hit the singularity point.

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

Â 7:30

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

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

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

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

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

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

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

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

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

Â projection point on.

Â 8:11

And the segments go singular.

Â You can see them going around.

Â They all complete this is kind of a polar plopped.

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

Â the magnitude is the magnitude of this cordinance.

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

Â 8:28

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

Â those coordinates go off to infinity.

Â Whereas the adus,

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

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

Â So that's kind of an illustration of that.

Â 9:13

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

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

Â something we have to look at.

Â So this is the attitude here, right?

Â You project it, here's your projection point.

Â This is my set of SSOPs.

Â 9:41

But this projection line between this set of coordinates,

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

Â I make a line with this projection point,

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

Â 10:03

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

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

Â a constrained altitude control, never exceeds 240 degrees,

Â 10:16

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

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

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

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

Â That's kind of the applications of these things.

Â So, as you get into this coordinate design,

Â quickly take a look at these issues.

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

Â which is a bad thing.

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

Â a completely different orientation.

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

Â 10:52

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

Â that were done.

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

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

Â And always being orthogonal centered in the origin.

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

Â the MRP behavior but also those asymmetric stereographic orientation parameters.

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

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

Â It's just this a parameter comes in and so

Â that you can write it on the very general way.

Â And so anyway also published the paper recently

Â about taking the surface and looking at different geographic

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

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

Â Different things that you can do with it, so

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

Â 11:54

Yes sir?

Â >> So you kind of mentioned optimizing

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

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

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

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

Â to be honest.

Â My primarily coordinates of use are MRPs.

Â I find that I can do anything.

Â I just have to deal with the discontinuity.

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

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

Â you just not going to tumble about this other axis,

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

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

Â go singular if you had to do that.

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

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

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

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

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

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

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

Â that you're still going to tumble past it.

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

Â 13:10

Life is full of details.

Â But you should be able to understand them at least.

Â Yes, Jordan.

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

Â maybe something with some curvature to it.

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

Â to make my life difficult, aren't you.

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

Â Ever is a big word.

Â >> Yeah.

Â 13:38

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

Â curvature to it.

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

Â You might have a different surface that then gives

Â you an even more linear behaviour.

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

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

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

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

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

Â Lots of fun little nuggets.

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

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

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

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

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

Â But there's an interesting world of kinematics and

Â how we can describe orientations.

Â So yeah, nice idea.

Â We should maybe explore that.

Â