21: MRP Form of the Cayley Transform

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From the course by University of Colorado Boulder

Kinematics: Describing the Motions of Spacecraft

51 ratings

University of Colorado Boulder

Kinematics: Describing the Motions of Spacecraft

51 ratings

Course 1 of 4 in the Specialization Spacecraft Dynamics and Control

The movement of bodies in space (like spacecraft, satellites, and space stations) must be predicted and controlled with precision in order to ensure safety and efficacy. Kinematics is a field that develops descriptions and predictions of the motion of these bodies in 3D space. This course in Kinematics covers four major topic areas: an introduction to particle kinematics, a deep dive into rigid body kinematics in two parts (starting with classic descriptions of motion using the directional cosine matrix and Euler angles, and concluding with a review of modern descriptors like quaternions and Classical and Modified Rodrigues parameters). The course ends with a look at static attitude determination, using modern algorithms to predict and execute relative orientations of bodies in space. After this course, you will be able to... * Differentiate a vector as seen by another rotating frame and derive frame dependent velocity and acceleration vectors * Apply the Transport Theorem to solve kinematic particle problems and translate between various sets of attitude descriptions * Add and subtract relative attitude descriptions and integrate those descriptions numerically to predict orientations over time * Derive the fundamental attitude coordinate properties of rigid bodies and determine attitude from a series of heading measurements

From the lesson

Rigid Body Kinematics II

This module covers modern attitude coordinate sets including Euler Parameters (quaternions), principal rotation parameters, Classical Rodrigues parameters, modified Rodrigues parameters, as well as stereographic orientation parameters. For each set the concepts of attitude addition and subtraction is developed, as well as mappings to other coordinate sets.

15: Modified Rodrigues Parameters Definitions9:29

16: MRP Stereographic Projection5:57

17: MRP Shadow Set Property7:58

18: MRP to DCM Relation4:27

19: MRP Addition and Subtraction4:54

20: MRP Differential Kinematic Equation14:10

21: MRP Form of the Cayley Transform7:47

Optional Review: MRP Definitions8:20

Optional Review: MRP Properties8:05

Meet the Instructors

Hanspeter Schaub
Glenn L. Murphy Chair of Engineering, Professor
Department of Aerospace Engineering Sciences

Now, we're going to go a little bit more quickly through this.

You've seen the Cayley Transform.

The Cayley Transform allows you to input an N dimensional,

skew-symmetric matrix and you put it into this math.

And outcomes in an orthogonal matrix, or what's really

awesome is you put the same math, instead of putting in a skew-symmetric matrix you

put in orthogonal matrix and with the same math outcomes skew-symmetric matrix.

So the forward mapping, inverse mapping was the same algebra,

it just depends on what the input is, which is kind of really cool.

So, way back in the day then, actually, I was close to wrapping my PhD.

[INAUDIBLE] and I we’re really pondering, can we expand this Cayley Theorem to,

the Cayley gave us basically for if this orthogonal is a three by three matrix,

proper orthogonal,

then the skew-symmetric matrix represents classical Rodriguez primers.

And then you can expand this concept of classical Rodriguez primers

to multi dimensional attitude descriptions which has applications and

structures in different kinds of fields.

We were wondering okay CRP's are nice but we know MRP's are so much cooler.

Let's face it right?

They're so much, lots of really nice properties that come with that.

Can we do that in higher dimensional spaces?

So this is what we found.

We can modify Caley's theorem without the squares here.

This is just one.

This is Cayley's theorem, and that takes a skew-symmetric matrix and

gives you orthogonal.

And the same mapping goes back again.

Here, the S's, as you will then guess, this becomes our sigmas.

This is a formula that will give, if you put in a skew-symmetric matrix of MRPs,

and put it into this math, you get to the DCM.

If it's a three dimensional space but

it turns out this property holds also for N dimensional spaces.

So we can use this to define higher dimensional MRP like coordinates that

define the orientation or, in this case, the state of a orthogonal matrix.

And dissolve in the system if I connect, this.

But we have to go to squares.

You can switch the order, that's nice, but one thing we do lose is,

with the classic Cayley, there was the inverse mapping with the same formula.

You just put the C in here, and outcomes an S in the front.

That no longer works once you go to higher dimensions.

And we're doing here too, there's a whole another paper we wrote together on

one with going to higher order, to nth order.

So we can do what's called higher order Rodriguez parameters instead of tangent

fee over four, we can do tangent fee over six, fee over ten.

But every order introduces more and

more singularities or, that you have to account for along the way.

So you flatten that curve, you get more and more linear responses, but

at the cost of increased number of singular points you have to account for.

But there's whole theories and papers on this.

So instead of doing the full rotation here, what we also looked at is

to do the inverse, basically we can take the square root of our orthogonal matrix.

Now what does that mean?

You get to four by having two times two, right.

Attitudes are matrix multiplication, so think of it that way.

So we have a matrix W that it squared becomes the DCM.

And if we multiply DCM's together so each W is a DCM essentially so

this rotation times the same rotation again, gives me the total rotation.

So the W the square root operator basically says look,

I'm giving you the half rotation DCM matrix.

And that's what that means geometrically.

Again, once in math, I'm not going to go through details.

I'm just trying to give some highlights of what people have looked at here.

So if you do this in the square root of diagonal,

this is a way that you can do it, matrix square root operation.

What it actually manifests itself as is we have this +1 because we know it's

a orthogonal matrix and the other one becomes complex conjugate pairs.

If it were three by three this is what you'd have as Eigenvalues.

There's that 1 + 1 we found in earlier work we did and

there's a complex conjugate set.

But this W now has angles over 2.

And if you do more dimensions, you don't just have one principle rotation angle,

if you go to higher dimensions, published a lot on this,

you have multiple principle angles.

It's like rotation supplemental folds and how does it all manifests.

And it really, you have to have some good bottle of Italian wine and anymore and

this will make a lot more sense.

If you're talking with these hands, kind of like what I'm doing and

it'll all look amazing.

But so just as we go to higher dimensional spaces, these ideas of principal rotation

angles expand but it doesn't just become one as we have a 3D but you get

multiple ones and is it not dimensional or even dimension affects all of this.

But you can do this, so now I can rewrite and go from the MRPs to this half

rotation, forward and backwards, using the classic Cayley formula.

And then there's all the symbols.

So then we regain all the properties we liked from the classic Cayley, but

you get this extra step of doing half rotations.

So that's been looked at.

And similarly, you can get your differential kinematic equations for

interdimensional stuff.

This is like a MRP rates and how it relates to these Omega's and

these whole theories and how to do all this stuff.

But if you interested you can look the formulas up and go there and to try and

create awareness.

And the math you learning and the projections you learning and

this principle rotation stuff.

We typically apply to three dimensional space but

this whole mathematical theories of taking these ideas.

To n-dimensional space as well, and

Cayley's theorem is kind of at the heart of that.

Jordan. >> I don't know if I missed this, but

if you go back one slide.

>> Yeah.

>> In this classical Cayley transform, is that S,

that's not the skew-symmetric MRPs, right?

>> This is the skew symmetric one But it's the formula,

it's like the Cayley, the classic.

And you can reverse the orders and

forward the mapping between W and S, it is just like regular Cayley.

>> Well, but W is the square root of DCM, right?

>> It's the half rotation, yes.

>> S the square root of- >> No, S is the full rotation.

>> Okay. >> And then this gives you this.

And then W squared gives you the actual rotation.

So, that's how the mathematics works out and you can go in Math Lab and

place some numbers, plug it in and you could prove this to yourself once you

see the pattern but I’ve kind of where those things go, good.

So there's lots of stuff actually,

because an open question was sent to me,

well about 10 years ago now.

Eigenvectors stuff from trying to program this off and

we've been using your formulas and doing this.

But they have structural components that are revolving.

So they are running into issues when they do this.

Why is that?

Well we know MRPs blow up if you don't switch them to shadow sense.

So an open research question, if any of you has too much spare time on

the weekends or the evenings is how do we actually switch from MRP sets shorts to

thew long for we can avoid singularities, how do we do this in higher dimensions?

Nobody has quite figured that part out yet.

The geometry, the mathematics of it.

There’s always new nuggets, if you kind of trying to thinking yourself to do or

curious, or if you work in this area.

And he was using some numerical tricks to do it, but

he was wondering there should be a nice analytical angle.

I agree. I just

haven't had time to delve more into this.

Definitely exists.

So all of this is ongoing work.

I'm just trying to show you some elements.

These things kind of come in chunks as somebody gets excited about this and

makes more stuff.

So unsolved problem.

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