0:05

Okay, moving on.

Â Deterministic attitude estimation, this is really about that at this instant

Â of time, I've taken at least two observations.

Â And for now we're just going to deal with two.

Â By Tuesday, we're going to talk about what if I have n?

Â What if I have four star trackers on there?

Â If the sponsors paid for four star trackers and you're only using two,

Â they get very pissed because they're like, why do we spend all this money, right?

Â So there has to be techniques that we can blend multiple things, as well.

Â And so this first one we'll do won't have that.

Â And that's called the vector triad method.

Â If you've taken 3200 with me you've probably seen this part already.

Â We'll take it a little bit further,

Â and then we'll get into all these newer methods.

Â So the triad method was actually quite popular.

Â There's all kinds of modifications of this one.

Â There's an end vector triad method out there and other kind of stuff.

Â But it's based on a really nice geometric principle and that means the triad frame.

Â We have two coordinate frames.

Â 1:02

When have n and we have b.

Â We have some inertial frame or some reference frame and then we have our body

Â frame, and then we try to find the bn matrix essentially, right, that's it.

Â But what we can do is instead of dealing with just two frames as often alive,

Â instead of just going straight forward from here two

Â observations I want my attitudes, sometimes it's easier to take a step back.

Â And we're doing that here mathematically by introducing a third frame,

Â by having this triad of frames, so we have three frames.

Â And this third frame we can define easily in terms of the measured quantities or we

Â can define easily in terms of the known, the n frame components of these quantities

Â and then we use matrix multiplication to tie them all together, all right.

Â So that's kind of in essence, that's the trick of the triad frame,

Â we go to a third frame.

Â So how was this frame defined?

Â We're saying this triad frame, I'm using t's, I have b, and here I'm using i

Â as the inertial frame, I'm introducing this new frame t, the third frame.

Â So the first access, you pick one of your observations and

Â immediately, Iâ€™m picking the s, there's a reason for that.

Â We talked about the uncertainties of magnetic field and

Â we talked about the uncertainties in the sun heading.

Â So Shayla,

Â which one is going to be more accurate, the m hat vector or the s hat vector?

Â 2:56

because now we're using this s hat information completely.

Â We're saying, okay, this is the sun heading.

Â I'm going to make a new frame that's going to line up my first axis

Â with the sun.

Â Then I have magnetic field, so what I do is my second axis is

Â orthogonal to the sun and orthogonal to the magnetic field.

Â So in this case I had a sun heading, that's t1, magnetic field and

Â it's a little bit hard to see in 3D but you do this one cross this one,

Â gives you a plus t2, right?

Â So we used the magnetic field to give me something orthogonal to the magnetic field

Â and orthogonal to the sun.

Â That's going to be the t2.

Â Once we have 2, the third is trivial.

Â t1 cross t2 has to give you t3.

Â It's a right handed coordinate frame.

Â If you do a left handed coordinate frame, I'm going to be really upset.

Â Okay, so that's how we define it.

Â 3:47

Now you notice this definition doesn't specify s is given an m or

Â n frame components.

Â It just says it's a vector.

Â The t1 vector is equal to this vector.

Â So we can actually define these frames two different ways.

Â Here's a quick 3D visualization if this helps, I'm not sure.

Â But t1, magnetic was this one, yellow was there.

Â So the sun was there, this crossed this.

Â This one here would be be my t2 then.

Â So that's the t1, t2 and then t3 is orthogonal.

Â Rotating maybe helps but it's just 3D.

Â Yes?

Â >> And what do we do if s and m are colinear in this case?

Â >> You are a trouble maker aren't you?

Â No, absolutely.

Â So, that's an important thing.

Â If somebody tells me I'm in 3D tumbling in a gyroscope in this room, and

Â Warda is to my right and you're to my right,

Â both of you guys are exactly lined up with me, that doesn't help me.

Â because there's still the infinity of orientations that I have.

Â So, that's where these methods breakdown.

Â There is no estimation technique that will take and colinear observations,

Â I'll give you a full 3D attitude.

Â You will only always get a 2D measure of the attitude.

Â You will not know the rotation about the.

Â Yeah, so that's a fundamental thing we have to make sure.

Â So it's also an estimation.

Â That's where rate gyros come in nice, because if you're doing magnetic field,

Â it's possible the magnetic field at some point might line up with the sun at that

Â instant and then it changes again.

Â So you have to account for that in your code, yep.

Â But for now, we are assuming then to non colinear observations.

Â Yes, so that was something usually think people afterwards, but that's good.

Â You're thinking ahead.

Â So this matrix math, or this vector math we did earlier,

Â I can do in matrix components by assigning coordinate frames.

Â We measure s in the d frame, and we know s in the n frame.

Â I know where I am, I know right now in the inertial frame, the sun was in that way.

Â That's it, so I can go through this same matrix math in inertial components and

Â in matrix components.

Â And in defining the t1, 2, 3s in two different ways.

Â And now if you go back and look at the basic DCM definitions,

Â when we had the BN matrix, remember?

Â The BN matrix, or the C matrix as we called it at the time,

Â the first row was nothing but b1 transpose, right?

Â It was something n1, something n2, something n3.

Â That is b1 in n frame components.

Â Or vice versa the first column was n1, n2, n3.

Â So here we have bt, so it's like bn then I had n1, n2, n3.

Â You did your homework on that on Homework 1.

Â Here we need it but instead of n we have t, change letters that's it.

Â So this is going to be t1 in b frame components, the second column is going to

Â be t2 and the third column is t3, all in b frame components.

Â For the nt matrix it's the same stuff.

Â Now the t1s in the n frame components are going to be the first,

Â second and third of this matrix.

Â So, that's kind of the beauty of the triad method.

Â You can now, from the measurement how we define this, we get the coordinate frame

Â express in n, the t axis expressed at n and b frame components.

Â So we could the estimate b relative to t and the inertial t.

Â And the final step is attitude addition.

Â Now I have three frames, right?

Â I know how to go from n to b and b to f, I need n to f directly.

Â That is nothing but matrix multiplication.

Â So I'm looking for the attitude of n to my estimated body, right,

Â that's what I'm trying to go after.

Â And it's going to be nt transpose, because that flips the order and then tb.

Â And that's it.

Â 8:07

And there's an infinity of n vectors that will give you the same orthogonal t2,

Â it just has to be kind of in the same plane.

Â So we don't use all of the information of n, that's why it's critical with the triad

Â method do the more accurate answer, make that your first axis.

Â because then we're going to part we're essentially only using half of

Â the information of the magnetic field sensor.

Â We need it in a 3D form but we're only using half of it mathematically.

Â So, it's nice, it's easy, it's quick.

Â You get something but you need to as an engineer you're going, you know,

Â I feel like leaving money on the table, right?

Â I have extra information that I'm not actually using.

Â There must be other ways that we can bind this and use everything.

Â And it'll be the following methods we get into here, but that is this method.

Â So any questions on the triad?

Â Many of you maybe had seen this before.

Â Hopefully every time you see it more and more will stick.

Â Spencer?

Â >> Most important part is that you take your most accurate and most reliable and

Â most trusted instrument measurement.

Â And that's what your first axis [INAUDIBLE]?

Â >> Yes, because then you can use all of the information.

Â Otherwise, you're throwing half of that one away and that's not a good strategy.

Â Yes, absolutely.

Â So for many missions, budget limited ones.

Â You don't all that kind of stuff.

Â This is very good and very simple and very fast,

Â especially if you don't understand full column filtering and other kind of things.

Â You could put this together, measure it very quickly, get some attitudes and

Â make things work.

Â This is actually flying on several different spacecraft for

Â sure simple as it is.

Â >> [INAUDIBLE] >> because

Â it's in a cross product form here.

Â So here s = t1 immediately.

Â I'm locking one of the axis there.

Â For the m, let's say m is just pointing towards the whiteboard for me, and

Â the first one is just pointing forward, make it simple.

Â That's one.

Â It makes no difference if m is pointing here, here, here.

Â As long as it's in this same plane there's an infinity of answers.

Â So with this cross product stuff we're actually losing some information.

Â We're not getting all of the end stuff that's there.

Â >> I see, okay.

Â >> All right, so that's where the mathematics,

Â that's where we're losing something.

Â But it's implied by being the second vector.

Â That's why it's really important make a better vector up here.

Â The second one is used to kind of.

Â Locking that, I know I'm pointing this way.

Â Now the question is how am I oriented and

Â if I go in my righthand side its pointing at you, it kind of locks it in.

Â But it does it as good as it can.

Â Spencer?

Â >> Seems kind of silly to use that kind of

Â 10:42

your most reliable measurement could be constantly coming from different censor.

Â So then. >> Why would they become

Â a different censor?

Â Lets say you have a magnetic field and a sun censor,

Â why would tumbling impact that?

Â >> If your sun censors pointing at 180 degrees away from the sun, then it.

Â >> You get multiple sun censors.

Â Yes?

Â It's not the sun's sensors fault you pointing it in the wrong direction.

Â In essence, spacecraft, especially with core sun sensor,

Â I've seen little cube sacs with 18 sensors on there.

Â My God, it's a wiring nightmare, getting all those things hooked up.

Â And as research going in, can we do it with less and less sensors and

Â have less ambiguities.

Â But yeah, so if you do this as a sun sensor source,

Â you'd have to have something else reliable that works no matter what.

Â And so they would make sure you've got four pies to radian coverage with that

Â sensor, absolutely, yeah.

Â So, there's a lot of things implied that you need to have

Â to make something like this work.

Â