0:05

So now we're going to talk about Modified Rodrigues Parameters.

Â These are quite popular, used in lots fo different areas.

Â Having just gone through

Â Rodrigues you will see that we can go pretty quickly here.

Â There's one subtle difference, but man,

Â this little difference makes a huge impact on how we can use them.

Â 0:40

If I have a 0 rotation Beta Nought is 1, so I divide it by 2, easy.

Â If I have upside down, Beta Nought goes to 0.

Â I still have 1, that's not bad.

Â So for what orientation now, will these go singular?

Â 0:56

Go ahead. [COUGH]

Â >> When it's upside down, 180?

Â >> No, 180 Beta Nought is 0.

Â >> So then 360, when it's- >> Yeah.

Â >> At 1. >> Okay, so wait a minute.

Â So if I'm telling you, this is the inertial frame, this is the body frame,

Â and they're identical, do you have a 0 or a 360 rotation?

Â So is the MRP singular or non-singular?

Â >> If you've done nothing, it would be zero, right?

Â Until you start cycling through.

Â >> Right, and how did you know it did nothing?

Â All you know was theyâ€™re identical.

Â So at this instant is my MRP description is singular or non-singular?

Â 1:43

Who is confused?

Â >> [LAUGH] >> Honest people.

Â Good! I hope you're a little bit confused.

Â That probably means you're thinking about this and go, wait a minute!

Â What's going on here?

Â This is where it starts to get a little bit weird.

Â But man it is great, it is really cool how we can take advantage of these things.

Â So, as you're very correctly summarizing,

Â you have to know the path, and if they're just identical right now,

Â you get to choose did it do one revolution or not, right?

Â And I typically say no, because who wants to deal with singularities, right?

Â So the singularity has been moved from 180 degrees all

Â the way back to 360 and 360 to same as the origin.

Â So that should give you a little bit of a headache until you kind of go

Â wait a minute.

Â Okay this starts to make sense.

Â So now the MRPs are like CRPs.

Â They're a ratio of the quaternion vector part over now one plus the scalar.

Â And that's going to shift things out and I can go very far in my rotations,

Â in fact I can do everything except for a complete revolution.

Â Tumbling upside down?

Â No problem.

Â I will just divide it by one, no big deal.

Â 2:48

Okay, so let's explore that.

Â As before, there's mappings, this is goes from quaternions to MRPs.

Â [COUGH] You can inverse this mapping using the quaternion constraint and

Â solve it and end up with these formulas.

Â This is one of your homeworks you'll do with this math and algebra.

Â It's this more ways to apply the quaternion constraint.

Â You need to get comfortable with that.

Â So there's nice elegant quaternion MRP direct mappings, definite singular,

Â every three parameter set by itself has to be singular somewhere, so

Â here is 360s unlike CRPS they blow up to infinity.

Â 3:23

And if you map it back with a little bit of a trigger to the [INAUDIBLE] you can

Â prove this now becomes tangent phi over 4, CRP was tangent phi over 2.

Â So, CRP, we could visualize small angles as being one,

Â two, three, your row pitch and young angles, essentially, right?

Â This linearizes to angled over 4.

Â So, whatever MRPs you have, you multiply it times four,

Â it's roughly that many radians.

Â Now how good is that approximation?

Â Tangent phi over four.

Â Let me see.

Â I was going to actually run this.

Â 4:17

If I go to well,

Â tangent phi over 2.

Â 180 degrees, I'm blowing up, I'm going to infinity, I go off the chart here, right.

Â That's what it act.

Â But you can see that I can be up to almost 50, 60 degrees and

Â there's barely any distinction between the linearized version which is tension x is

Â equal to x and the actual tangent function.

Â So this gives you a quick visual illustration that the CRPs actually

Â lineralized much better then the oiler angels.

Â Oiler angels, 60 degrees, you better have some high order terms in there.

Â 4:54

Now, if we go here to tangent phi over 4.

Â Look what happens.

Â Really, I mean it's almost linear up to 90 hundred-ish degrees that you can do.

Â If you're going from zero to 90, 100,

Â that's a huge domain that you can deal in a linear if you do a linearized analysis,

Â linear feedback gain analysis which we'll get to.

Â This was actually really valid and if you exceed that domain,

Â if you look at this curve, keys wise, it's almost straight, right?

Â Yes, it's starts to deviate from the blue line but, it's not taking off to infinity,

Â that crazy, all right?

Â So, this is how well these things linearized,

Â they're really really good for large rotations and

Â the linearized equations are quite representative of the non-linear response.

Â They're not exact but you get a very good estimate, which is going to be very,

Â very handy for feedback gain analysis.

Â We can go higher order, actually with a Cayley transform.

Â I've got whole papers published on this, with Hawkins and

Â Jenkins, almost called Higher Order Rodriguez Parameters.

Â It turns out we can take these things to a higher order in the Cayley transform and

Â I can come up with coordinates that are tangent fee over six,

Â tangent fee over eight.

Â And you can arbitrarily bring this closer and closer but you're dealing with more

Â local singularities as you go to higher stoppers.

Â You have six sets of coordinates but

Â six singular conditions we have to keep switching between them so

Â things get a little more, for every gain there's a price.

Â But, you can expand this, so

Â here you see the tangent phi over four part and what happens there.

Â 6:32

Okay and there's also, if you can do the math,

Â there's ways to relate these CRPs to MRPs, and we can go back and forth.

Â So all these Rodriguez tend to have nice compact direct relationships.

Â We dont have to go to and from DCMs all the time.

Â Just to relate one for the other.

Â DCMs work, but it's nice when you got these really compact, elegant,

Â analytical answers.

Â So modified Rodrigues parameters.

Â We have 1 plus this.

Â That moves it as far out.

Â 7:12

It is real reasons for that all over a sudden,

Â you could do that, then you have a set that never go singular,

Â but if I give you an attitude in that set,

Â those attitude coordinates can all of a sudden represent two different attitudes.

Â So if I give you an MRP of one,

Â two, three, you wouldn't actually know what the attitude is.

Â There's two possible conditions, which is hell.

Â [LAUGH] If you're doing control estimation, then one's a true attitude and

Â one is of wrong attitude.

Â And you don't know which is which.

Â So, that's why one turn out is the best we can do.

Â And you'll see a geometric reason for

Â that shortly as we look at the stereo graphic projection.

Â 7:58

>> But if you have gone to 190 degrees and you want to find the short way back to

Â zero for controlling, it was easy with the because you just.

Â >> Yes. >> The negative.

Â How do you do it with these?

Â >> I'll get to that.

Â We are getting there so this is just different ways to go from the DCM to here,

Â like with the CRPs, there's some kind of back formulas so if you're coding them.

Â But they all will have issues and now if you're doing a 360 rotation as along as

Â Beta Naught is not minus one then you are fine, so this actually works pretty well.

Â To get one of them, the short rotation, in fact if DCM,

Â as long you pick the Beta Naught that is positive,

Â the positive square root of this you're guaranteed to have the short rotation.

Â So this would actually be a non-singular way to always get to short MRP back,

Â what we will see is there are, CRPs were unique.

Â 8:52

That CRP and the shadow set on that sphere,

Â both those points projected to the same point in the cube space.

Â MRP's those two points will project to different locations.

Â So there's two possible MRP sets.

Â And they're very equivalent to quaternions.

Â One is short, one is long.

Â This math will actually always give you the short one.

Â If you want 80, there's an ambiguity, but pick one, they're both equally good.

Â All right, if you 180 down or left or right, it doesn't really matter.

Â But the mathematics work out just fine around that.

Â