0:18

>> Unit vector.

Â >> A unit vector. Okay, that's a good answer.

Â So, how may independent degrees of freedom do you measure with a unit vector?

Â >> [INAUDIBLE].

Â >> Two.

Â But a unit vector consist of, three coordinates.

Â You can measure it relative to a body.

Â There's an X, Y and Z coordinate.

Â That's the direction the sun was.

Â This is the direction magnetic field is, right.

Â So while we get three coordinates, it's just a unit vector.

Â So through the unit constraint, there's only two that are really independent.

Â You can also think of it as [INAUDIBLE] elevation.

Â You're really just getting two angles, gives you a heading, but

Â we took an expressive as a three by one vector.

Â That's typically what you have.

Â So, that's the observation.

Â So let's some highlights.

Â 0:59

There are several assumptions to do this estimation.

Â We assumed a, and b, and c, and a bunch of things.

Â So, to give you some highlights.

Â What did we assume with this estimation problem?

Â >> [INAUDIBLE] >> Okay.

Â In this room, if you think of this, you do know if I'm the one doing the estimations,

Â and somebody spun me around.

Â I need to know where in the room am I?

Â Otherwise saying hey, this big whiteboard is to my left, means one thing for

Â my orientation, If I'm in a different location in the room,

Â that thing being to my left, I'm actually pointing in a very different way.

Â So that means for us to know where in the orbit am I,

Â or enough information to resolve that.

Â What else?

Â Known location.

Â 1:44

CK.

Â >> [INAUDIBLE] >> Environment, you need to know.

Â This count for zen thing, Kung fu fighting or something.

Â Know your environment, all right.

Â What was that?

Â On Batman, the first one with Liam Niesen,

Â when he's teaching him make some break through the ice, all of that.

Â Know your environment stuff, all right.

Â It's that kind of a thing, but it's important, because otherwise if somebody

Â just tells me hey, you're in this room and you're here.

Â And the white board's to my left.

Â 2:13

And you guys shuffle the white board all over the place,

Â I have no idea where I'm actually pointing.

Â I need to know where is the white board in this room to be able to do that.

Â For us, that means we need to know where is the sun right now in space,

Â relative to my location?

Â What is the magnetic field?

Â If I don't know the magnetic field, I can measure it, but

Â I can't do much with it, right?

Â So, all of these things are important.

Â So, those are some key stuff.

Â Now how does estimation work, fundamentally?

Â What do we measure, one else do we need to make this completed?

Â 2:59

You remember something, Andrei?

Â >> [INAUDIBLE] >> Okay, so let me just make a V1.

Â >> [COUGH] >> I'd say that's my observations we're

Â talking about, right?

Â Something that I'm measuring.

Â >> [COUGH] >> You need to know it in which space?

Â 3:21

>> [INAUDIBLE] >> If you wanted to do

Â a different problem, let us say I am doing relative heading.

Â I am using a camera, you have all seen this.

Â Like facial trackers that track features on a spacecraft.

Â Each gives you a heading.

Â I know the space craft, I know the geometry,

Â I know my location trying to figure out what's my heading.

Â This is kind of classic vision problems I got too.

Â You may not be measuring relative to an inertial frame, right.

Â You might be doing, so here in class we typically treat it as an inertial.

Â But it's not a fundamental requirement of this estimation technique that we have.

Â There's no here, it's really a purely kinematic relation of

Â how do these headings expressed in one frame relate to another frame.

Â So good, so I'm glad you said that.

Â Inertia is typically what we use, but just remember it's not required to be inertial,

Â this could be something else as well.

Â Good, and then we have the measurement which is always in the body frame, right?

Â That's what we care about with spacecraft.

Â In the body you have the magnetometer, or sun sensors, horizon sensors.

Â Star trackers, things that give me different kind of headings.

Â Visual sensors, cameras, all this kind of stuff.

Â And the question then is how does this relate?

Â 4:43

Now I'm going to add a bar over stuff, because this is what we're trying to find.

Â I won't get, unless I'm really, really lucky, the true body attitude.

Â But I tend to get an estimated version of it.

Â So this is how I differentiate in these notations between,

Â this is my estimated body.

Â Versus the truth body.

Â Then we can look at B bar to find the actual estimation errors.

Â This is it.

Â Will I be able to use one observation, to do the complete general attitude problem.

Â Trevor? >> [INAUDIBLE]

Â >> Okay, so one,

Â as we were talking about earlier,

Â mentioned this was only two degrees of freedom.

Â Attitude, is a three degrees of freedom problem.

Â So one is not enough.

Â It's immediately under-determined.

Â If you do two measurements, it's immediately over-determined.

Â And that's just a fundamental issue in life.

Â [LAUGH] And it would be nice if we had one that was just,

Â what we needed to keep it simple and just move it on.

Â It's not quite that way, right.

Â So that's a fundamental thing that we have.

Â So good, we need a second measurement using all unit vectors.

Â So, I'm just going to have the hat in the B frame.

Â And it's the same estimated attitudes that will map one to another.

Â 6:09

>> [LAUGH] >> [INAUDIBLE]

Â >> Okay, where I lose you?

Â Step one?

Â >> [INAUDIBLE] >> [LAUGH] Okay.

Â So we'll let you recover.

Â Okay, so Nick, help that out, will there be a single dcm that

Â perfectly maps these known quantities into these measured quantities?

Â >> No.

Â >> Why not? >> [INAUDIBLE]

Â >> Which one?

Â >> [INAUDIBLE] I don't really know.

Â I also realized in this last lecture one

Â I sat down [INAUDIBLE] >> Definitely, catch up.

Â 6:57

Will those areas in there, right?

Â So the answer was correct, no, you don't get it, right?

Â But this is where I'm trying to review quickly fundamentally when you do this

Â homework make sure you're on the right track.

Â because, some people go, well, I tried this and

Â I can match one of my vectors, but the other one's not right.

Â Well, that's probably still correct.

Â You won't be able to.

Â because, with measurement noise there could be noise in here.

Â Maybe you can find a DCM that perfectly matches one vector into another.

Â But then it's not going to match the other.

Â 7:24

So it's kind of like, I'm really critical on this one, but this one I'm just not

Â going to be able to match well, because with noise it won't always be the same.

Â Is like when you do classic highschool physics problems, and

Â you have to use voltage versus current measurements on the resistor or something.

Â It's supposed to be a straight line, but who ever got a straight line?

Â I never did. The line went all over the place, right?

Â because that's real life, it's measurements, noise Corruptions.

Â You can have this.

Â So, it won't be one.

Â So immediately we have to recognize with estimation,

Â life is going to be a little bit more difficult.

Â What is the best that we pick, right?

Â And there's different ways to do this and that's what we're going to do.

Â Now there is different ways to define best.

Â 8:03

There was one lady who defined the formula.

Â Anybody remember her name?

Â Wahba, right?

Â She came up, it's the classic thing.

Â If you look up Wahba's problem, tons of publications on how this could be solved.

Â But they all solve the same optimality problem.

Â And we saw last time, well, it didn't solve Wahba's problem actually.

Â 8:22

Which one was that, which estimation method did we do?

Â Andrew?

Â >> Triad. >> The triad method,

Â right, the triad method doesn't solve an optimality problem.

Â Who can outline for me what the triad method did?

Â Just in basic words.

Â 8:38

Go ahead Jordan.

Â >> You pick your best [INAUDIBLE] >> But our best measurement,

Â we typically said that was the sun, if you had sun and magnetic.

Â It was very, sun's tend to be way more accurate than magnetic fields.

Â There's a lot of uncertainty in magnetic fields, right?

Â Good, so we set that one.

Â How do we get our second axis?

Â >> Cross it into your second measurement and

Â normalize >> Crossed with the magnetic field.

Â And this is normalized by itself to make a unit vector, that should be a 2.

Â And then 3 is t1 crossed to t2 which gives you a right handed

Â coordinate frame, right.

Â So we define, instead of going, I want the additive between these two frames,

Â we actually step back and say well, It's easier to get the attitudes of body and

Â inertial relative to a third independent frame, a different frame.

Â And then we reassemble it.

Â And that's the trick.

Â So we have measurements in the N frame, so we can do all this stuff in the N frame,

Â and in the end, you can get TN and

Â we have everything in the B frame, so DN, we can get BN, no,

Â BT, not BN, just messed it up.

Â 10:03

And then you multiply them out with the transpose, and that will give you BN.

Â I mean, that's fundamentally,

Â in quick terms, what the triad method to do this right.

Â So, there's no optimality.

Â We put in this light waving saying which one is better.

Â That means we use one of the vectors completely, setting t one equal to s hat.

Â I'm using that all information.

Â All those two degrees of freedom we're talking about.

Â And the m, we're just using partially

Â to get something that's orthogonal >> [COUGH]

Â >> To this and this but there's

Â an infinity of m's that would actually give you that mathematically, right?

Â So, that's it, but I cannot say the s is ten times better than m.

Â 10:39

To use it more or maybe the m is only part,

Â you know slightly worse then this sensor because it's a very coarse sensor.

Â There's no way to put weights into this function.

Â One of them is picked completely, the other one is picked partially.

Â That's all the knobs you have.

Â And then, you just run through it and you get it.

Â And there's different versions of this, right.

Â Whereas [INAUDIBLE] problem, really formulates the estimation problem

Â as a least squares optimization, where you're saying okay,

Â this in the body frame minus as Robert was saying,

Â right, we're looking for this DCN that maps the known quantity,

Â I'll add indices in front of this, in the inertial frame.

Â This will give you your error, right.

Â This is supposed to be exactly equal to this, but we know with noise, corruptions,

Â uncertainties of where I'm really at there's always these error sources.

Â This is not going to give me perfectly a zero vector.

Â But I want to make him as small as possible, in a least square sets.

Â So you do this basically and you would, in matrix math, you would transpose.

Â Vi-hat B- the estimated Vi-hat N, that's it.

Â And now, we have N of these.

Â So there's a cos function that sums up over, wait, equal to 1 to N.

Â 12:34

>> Higher weight would be for a better measurement.

Â >> Right, does the absolute value matter?

Â >> No.

Â >> That's the key thing to remember.

Â Some people go, you didn't tell me what weights to use.

Â I just know they're equally g >> [COUGH]

Â >> Well if they're equally good make them

Â ten and ten, make them one and one.

Â People often like weights around one just as [INAUDIBLE] condition but.

Â That's it.

Â So the relative magnitude is important.

Â The absolute magnitude, doesn't actually matter in this, okay?

Â 13:05

Now, let's look at some other methods.

Â So we went through the triad method >> [SOUND]

Â >> And that was good.

Â We get an answer, but there's no way to add weights.

Â There's no way, this method, to add five observations.

Â I only can use two, in fact, actually I used one and a half.

Â That's it. [INAUDIBLE] you had a question for me?

Â >> In this problem, are we trying to [INAUDIBLE]

Â cost function?

Â >> Yes.

Â In Wahba's formulation, we're trying to minimize this cost function J,

Â because that means this vector ideally should be equal to this matrix, right.

Â These matrices have to be the same, if I have no measurement [INAUDIBLE] and

Â everything's perfectly lined up, but in real life with all these corruptions and

Â uncertainty, it's not going to happen.

Â So we want to make this residual, this is your least squares residual,

Â as small as possible, so that finds that optimal fit.

Â Which is kind of like this, kind of problem which is y equal to ax plus b.

Â But it's done in a 3D attitude sense, which makes it look more complicated.

Â But I'll be jumping back to this as an analogy.

Â Several times as we go through this.

Â