0:05

I'm just going to step through this, we did it once before for the triad.

Â Let me make sure this is all initialized.

Â So the setup is the same thing, we setup the truth.

Â I'm setting softs and actual attitude, I'm getting, these are my known inertial

Â directions, and then I'm mapping the inertial directions into the body frames.

Â So these are my truth observations.

Â And then I simply setup measured states

Â 0:32

which are slightly corrupted versions of that.

Â So you can see truth would've been 813 something.

Â I'm giving 819, I just manually made it.

Â You can throw in random noise or something too.

Â Yes, sir? >> What are the weaknesses of

Â the Davenport Q Method?

Â You said there was a different algorithm that applies in a lot of stuff.

Â >> Okay, let's talk about that now.

Â 1:17

The computer will do it, and it does it pretty quickly for a three by three.

Â If you do this for a four by four, just time Matlab once.

Â Do the timing operation.

Â Do it on a general three by three you've randomly generated,

Â then do it on a four by four.

Â Then do it on the five by part.

Â 1:32

For every dimension extra, it's not just 10% more time.

Â It's like a quantum jump in hour much time it takes to evaluate it.

Â So the biggest challenge, especially many years ago when we weren't flying very fast

Â processors in space, we were worried about radiation hardening.

Â As well as with the latest Pentium chips that were flying, or Intel chips.

Â You worry about computational effort.

Â And solving a four by four eigenvalue eigenvector problem on a flight computer

Â can be a big challenge to do this.

Â 1:59

So, when we get to the next method quest, they all solve problem,

Â they give you the same answer to within numerical precision.

Â The differences in all these research papers analysis is how fast I can get it.

Â And the is a very fast one.

Â There's one that's called ESOQ, Dariele Morari did, ESOQ 1 and 2.

Â He showed they're actually a little bit faster than they even.

Â But there's lots of different formulations that have been done out there, and

Â it's all about speed, how quickly we can get to this answer.

Â So no, great question.

Â So this was just a setup, right?

Â Same as the triad.

Â I'm not going to spend much time with this.

Â In fact, it's the same corruption.

Â That's why I didn't apply random numbers.

Â I want to make sure I'm using the same corrupted measurements across all

Â the different methods.

Â So now, how do we do this.

Â In this problem, I'm just saying look I'm setting the weights equal.

Â So I'm not making any distinction into accuracy in one sense or to another.

Â The triad just used one completely and kind of partially used the other.

Â Here I'm going to use both equally in this measure.

Â Now I go ahead and

Â 2:59

I go I evaluate this b matrix which you can see is a three by three.

Â B plus the transpose gets you the s, the trace gets you sigma.

Â Then I composed the z and in the end I put this all together.

Â And you come up with a four by four, right?

Â There's a homework problem that has you do exactly this stuff just with different

Â measurements, different kinds of states.

Â A little bit more challenges in there.

Â So that's it.

Â Now we've got my keg matrix.

Â Next thing I do is I evaluate the mathematic that's called Igon's system.

Â So here's my four Igon values that I have.

Â So which one is the biggest?

Â 3:35

Clearly, this one.

Â Plus 1.99.

Â That's minus.

Â And then it's 0.03 the first one is clearly the biggest.

Â And, so, I am going to set my perconia set equal to

Â the first eigenvector because that was the biggest one.

Â And, the answer is already unit length.

Â That's just what these algorithms happen to give us.

Â Now, here, you can see the scalar part is positive.

Â Which means this is actually the short rotation of that attitude measure.

Â If this was flipped and you cared about it,

Â you could simply reverse the sign of all four of them again, right.

Â But that's how you find four possible ways.

Â Now let's see how good it is in estimation.

Â And if I know simply map this quarternine set into EP to C.

Â Then mapping it to the direction cosine matrix, I get this direction cosine matrix

Â and like with the triad method, we have the true attitude.

Â That's how we set up this problem.

Â I'm going to do b bar n transposed with b and true.

Â That will give me b bar relative to b so

Â that's the attitude of the estimated body frame relative to the true body frame.

Â Where is that?

Â And then I pull out the principal rotation angle and that's what I'm doing here.

Â So, as expected the DCM of b of the estimated body relative to the true body

Â is almost identity.

Â Which was what we hoped for, right.

Â If it were identity, it would be 0 estimation error.

Â And you're really, really lucky.

Â But, it should be something hopefully close to that.

Â And then you pull out the added actual angle error and it's about 1.69 degrees.

Â And if you compare the same math with the triad method.

Â You may not remember it.

Â But, that was somewhere I would have to evaluate all of those.

Â 1.85, right.

Â So, actually we did better which is nice.

Â We used all of the first measurement and the second.

Â So let's, I did better.

Â That sounds great.

Â Are you guaranteed that the triad will do worse

Â 5:29

than did Davenport's Q method, right?

Â Davenport Q method has to work optimally in there.

Â We're optimizing a cost function, and

Â that's a great sales pitch when you're talking to sponsors and stuff.

Â This one's going to be optimal.

Â But be very careful, are you sure that optimal is better than triad?

Â What do you think?

Â >> Well, you chose the waits, so if you chose the wrong measurement to wait.

Â Could be worst.

Â Even LQR control people go, awe, it's optimal controlled.

Â Yes, but it's with randomly human chosen weights.

Â So, other weights may be do different things.

Â That's one part of it.

Â But, the other thing is you're using both parts.

Â But, let's some some sensor you were using, it was perfect.

Â 6:21

In that case, the triad, I mean, it's with random numbers.

Â This is unlikely but this could happen.

Â So if you run lots of Monte Carlos, you may have a situation where

Â the triad method uses all of this one that happens to be really, really good and

Â part of this one, and using the parts that are good.

Â The other parts we happen to ignore.

Â It's mathematically possible for

Â the triad method to give you an answer that's better than just optimal method.

Â 6:44

It's possible.

Â Is it likely?

Â No.

Â [LAUGH] All right.

Â If we run enough runs,

Â that's a very we have to be very lucky in how these errors have lined up.

Â But if you are running enough runs and if you're going a demo to a sponsor,

Â the demo gods kick in, and you're very likely to have a random

Â number that's going to give you something where try it is better.

Â And you're trying to prove to the sponsor, hey, your cool new algorithm is better.

Â Not necessarily.

Â So, run it statistically.

Â Run it a hundred times, a thousand times.

Â That kind of stuff.

Â This is where you get a good measure and you can show hey, statistically,

Â Davenport should beat Triad but in any instance that may or may not be true.

Â