0:15

And specifically, in this discussion,

Â we're going to be doing it with instantaneous measurements.

Â And from that we have to resolve.

Â I'm not taking one, the rate gyro thing here, and

Â then one star tracker measurement there.

Â You have to blend it, then you have to go to filtering techniques like common

Â filters or unscented filters.

Â Or there's least squares filters or other kinds of filters out there, right?

Â And for those of you who've taken estimation or orbit determination,

Â you've realized that's at least a class worth of stuff just to get into

Â covariances, and uncertainties, and all this stuff.

Â We're not dealing with that in this class.

Â So this class is all dealing about instantaneous stuff.

Â I have an observation.

Â The sun is here and magnetic field is doing this.

Â So the moon is there, and my local horizon is this.

Â I have different observations, and we'll define what we mean by that.

Â And at this instant, how do I resolve what my orientation must be?

Â And how do I deal in, particularly, wave undersensed situations,

Â oversensed situations?

Â How do you make it optimal?

Â That's Walbach's problem.

Â We'll be going through her formulation on how to describe the mathematics of

Â how good these orientations are.

Â And we'll be using a lot of the kinematics you've just seen.

Â Especially how to do different orientations and how good is the measure.

Â We'll be using the principal rotation angle.

Â It's a convenient thing that just says, yep, that we're good to within one degree.

Â But that's kind of the high level.

Â We'll cover it today, we'll wrap this up probably on Tuesday.

Â And your next assignments actually include some of these methods as well.

Â 1:42

Attitude Determination, so static attitude determination,

Â that's what we're doing in class.

Â Basically, it's all instances.

Â Somehow you're getting complete 3D headings, or you're getting just partials.

Â But we're never getting just rates.

Â Forget rate gyro, sense of rates, star tracker rates.

Â Then you're really looking at an integrated system, a filtering system,

Â that's a whole other class, right?

Â And that's an important topic, just not something that we are covering in here.

Â Because not everybody here has had estimation theory.

Â So, that's a whole other thing.

Â But what with I am giving you here, this becomes the input to a common filter.

Â At some point, you go, hey,

Â I have a new observation, if you have dealt with a common filter before.

Â This is how you get to that new observation.

Â And there are some classic formulas that you should be aware of.

Â So the dynamic is something, common filter, rate based, those kind of things.

Â That's a whole other thing.

Â 2:34

Basic concept, how many direction vectors do we need in 2D?

Â So that's basically planar motion, fixed axis rotation,

Â that's what I'm talking about here.

Â In this room, if I'm just blindfolded, somebody spins me like crazy,

Â okay, and how many chunks of information do I need?

Â Do I need to know where the whiteboards are, the projection boards,

Â and the actual door to where you guys are sitting?

Â How many chunks would actually tell me uniquely how I'm oriented?

Â If you're doing fixed axis rotation.

Â Evan, what do you think?

Â >> I want to say two.

Â >> Okay, give me an example.

Â >> An angle for you to turn to face that object.

Â And then a magnitude to go to it.

Â >> But the angle is a magnitude.

Â But to think simpler, if somebody spins you, all right.

Â 3:53

>> Yes, but we're looking at fixed axis rotation.

Â So if I know my orientation, no, that's a good point.

Â You could be upside down, it still does it.

Â Actually, there's multiple then if I'm allowing other.

Â But let's say I know I'm standing upright.

Â That means something.

Â Yes, sir.

Â >> You know your position.

Â >> You know your position actually because that's a really important detail isn't it?

Â I'm standing here and saying hints to my right, clearly I'm facing you guys but

Â what if I'm standing up here and I say the desk is to my right?

Â Well, all the sudden I have a different orientation.

Â 4:55

one chunk of information, that's all I need if somebody tells me that whiteboard

Â is right straight ahead, I must be pointing this way in the room, right.

Â Also requires knowledge of what this room looks like.

Â If I take David, yes.

Â >> Daniel.

Â >> Daniel, man.

Â As soon as I think about it, I'm always getting them off.

Â Daniel, so if I take Daniel, blind fold you, take you in an arbitrary room at CU,

Â spin you around and say, hey, you're facing the whiteboard,

Â how are you pointing?

Â Any chance of knowing how you're going to do that?

Â >> Yeah, coordinate thing.

Â >> Yeah, you need to know your environment so besides knowing where you are in that

Â environment, you need to so, hey, this is the solar system.

Â This is how where the suns are aligned up with us, right?

Â That's where those stars.

Â If I know I'm pointing out Polaris, great, but which galaxy am I basically?

Â That's what you'd have to know, hopefully we know where we are.

Â But you never know.

Â There is all kinds of movies worm holes,

Â who know where we'll be going ten years from now.

Â So, we need to know the environment, we need to know where we are so

Â there is a lot of key assumption that can go into this estimation theory and

Â what we're going to break down is if somebody tells you something like hey,

Â something is to my right, it's straight ahead it's up into the left.

Â How do we break this down mathematically and compose a full attitude measure?

Â Yes? >> So

Â we're just concerned with orientation and not so much location.

Â >> Yes, we're assuming we know location already.

Â Yes, absolutely.

Â 6:24

So for us, it's basically like a compass.

Â That's just one information, as soon as I'm telling you you're facing North,

Â that's one chunk of information.

Â That's assumes you know you're on the Earth,

Â you know where on the Earth you are.

Â You know that the north means it points upward,

Â that means you know your local environment.

Â I now know how I am pointed, but it assumes a lot of knowns.

Â So it kind of, as you do estimations and locations, keep that in mind,

Â that's always implicit.

Â If we do three dimensional motions and

Â this is kind of where Kaley's question comes in.

Â 7:30

It's still to my right, right?

Â Or I can't go upside down.

Â I can't do a handstand, not without looking really silly.

Â Right, there's actually a whole infinity of ways that you can rotate about this

Â axis and that is always to your right, because we're looking at a single axis.

Â I'm not doing recognition going I am right side up to that port or something,

Â we don't have that, it's just a single heading, these are all heading

Â informations and headings are fundamentally a unit direction vector.

Â So while you may get three coordinates for that, because of the norm constraint,

Â it's really two chunks of information.

Â So one heading information that says,

Â hey, this relative to the body, it's that other object, is in this direction.

Â Or the magnetic field, or whatever you're measuring, is in this direction.

Â That gives you an axis.

Â That's what a heading is, it's not full 3D measure.

Â 9:02

How many coordinates do you need, at least, to define your attitude?

Â Three.

Â So, one heading vector, gives me two chunks of information.

Â Two degrees of freedom.

Â Think of it as azimuth elevation if you use spherical coordinates.

Â That's one way to think of mathematics, right?

Â That gives you two.

Â So if I add a second heading, I go from two to four.

Â And that's kind of at the cracks of attitude estimation.

Â Either I have too little, do it in 3D or immediately I have too much.

Â And if I have too much, well do I throw something away?

Â Do I blend it and if I blend it, how do I blend it?

Â So what you get in attitudes are always unidirectional vectors and

Â it's in chunks of two essentially, as a unit vector.

Â And that's a problem.

Â 10:17

So I'm going to now s and m, just as defaults, they are nice,

Â they are very convenient, small size use them a lot.

Â A core sun sensor that will tell you roughly where the sun is for

Â this body and would be a magnetic field sensor

Â that will tell you what the magnetic field is on this body.

Â It assumes that you actually know where the sun is, so

Â I don't need to know actually where we are on Earth orbit.

Â I just need to know what Julian date we have, so where's the Earth around the sun.

Â That's precise enough, you can then figure out.

Â Okay, right now if I have that orientation,

Â this is where that initial vector would be.

Â So we could know that.

Â Magnetic field means if you want to reconstruct it,

Â you need to know what is the magnetic field as seen by the inertial frame.

Â Fundamentally, I'm trying to estimate what is body orientation

Â relative to inertial frame.

Â So for now let's just use the ECI frame, earth centered inertial frame.

Â It's a non rotating frame centered at the middle of the earth, that's it.

Â So the magnetic field, we need to know at this instant how far has the earth

Â rotated, what's the magnetic field doing?

Â And as you imagine, a lot of uncertainty with that as well.

Â But you need that information.

Â because otherwise, this room could be morphing and

Â I have no idea if the door is not over here or here.

Â I need to know what the environment is.

Â The same thing with the sun.

Â So you take your measurements in the body frame.

Â So I get these quantities.

Â But as seen by the body, the sun is 0.1, 0.2, point something in that direction and

Â the minute the field is measured, it's a different vector, right.

Â I have to know where are they in the environment as seen by the notion frame

Â and this will allow me now to come up with my estimated attitude.

Â So I'm using a b bar here to kind of denote that.

Â B is typically the true body frame.

Â And the estimated frame won't be, unless you are very lucky,

Â exactly the same as the true body frame, right?

Â So, this is what we want to estimate.

Â What's the rotation matrix that will map these known quantities?

Â I know where I am,

Â here over the pole this is what the magnetic field should be doing.

Â That's the vector in M frame components, right?

Â If I knew my attitude, I could map this into body frame components.

Â And those mapped coordinates and the measured coordinates better be the same,

Â otherwise you don't have the right DCM.

Â I mean no one of them he can do that with two of them there's ways to do it.

Â And we'll look at different ways this can happen.

Â But that's essentially the thing.

Â How do we find this attitude matrix going from n to b?

Â The bn matrix, that's what we're after.

Â How do we reconstruct this?

Â Yes, sir.

Â >> Say that B bar, what was that again?

Â >> This is the estimated attitude, right.

Â Ideally, there would be only one and this is it and then we're set but.

Â >> [COUGH] >> In real life,

Â you have measurement errors.

Â We have all kinds of sources of errors.

Â We'll go through those in a moment as well.

Â 13:06

Actually let's just do this now.

Â What sources of errors do we have?

Â Let me try to estimate this.

Â Well the easy ones, I'll start off there, the measurements themselves.

Â You might have electrical noise.

Â You might have digital noise like you only have an 8 bit converter

Â maybe a 12 bit converter that's truncation errors you have deal with, right?

Â So the measurement clearly will have noise and issues always.

Â There's never perfect.

Â Where else do we have noise in here or issues?

Â >> The magnetic field is dynamic.

Â >> Yeah, knowing this field.

Â >> This part.

Â 14:05

>> The Sun is pretty good.

Â We know where, but, let me see.

Â If we draw this out, this is not to scale.

Â There is your Sun, right, we're way out, here's earth,

Â here's your satellite and it's orbiting.

Â This distance is miniscule compared to the 149 million kilometers, sun,

Â earth distance is, roughly, all right.

Â But still, you could account for all of that.

Â But, how precisely do we know where the earth is around the sun?

Â Well, we know it actually really well.

Â But not to infinite precision.

Â And then if you want to account for

Â the satellite's motion, how good is your orbit determination?

Â If you want to account for those small error that you actually, not just moving

Â around the sun with the earth, but you're kind of wiggling around it.

Â If you want to account for that, how good is your orbit determination?

Â It immediately comes in.

Â So these are all error sources that you'd have to put into a complete analysis and

Â account for.

Â For now, I'm just saying we assume we know it, we recognize as error sources,

Â but in this mathematical step, there's nothing we can do about it yet.

Â So this is fundamentally.

Â We measure this, we need to know our environment.

Â So the unit vectors we're measuring, we have to know in our environment

Â in inertial then there's mathematics to come up with an attitude measure.

Â Sometimes it'll be a DCM, it'll be sometimes quaternions, it'll be also CRPs.

Â The different methods have different mathematics on how to reconstruct this.

Â Yes.

Â >> So, since you have measured and given for both N hat S hat.

Â Don't you only need one of them to get your?

Â >> No, you still need two, because if I only have one, really geometrically

Â it means I know you're straight ahead of me, but in the space station,

Â I could have an infinity of attitudes and have you straight ahead of me.

Â The math here you can't just take this and

Â invert this somehow and reconstruct the full three by three matrix.

Â >> Right, so the measured and the given,

Â you need two for both of those because then those give you your two frames.

Â >> Yeah, so from this I need to know that, and

Â I need to know where each one is given in the environment.

Â If I didn't know where you are in this room and something just tells me,

Â hey, you're right ahead of me, I can't tell where I'm going to go, right.

Â So for every measurement, you need to know this part as well, right?

Â So it comes back to the basic stuff we were actually very familiar

Â with from everyday life, we just have to expand it from 3D motion in space.

Â