0:05

>> Now, we're going to do this example here.

Â So, we've got a spacecraft flying out there.

Â Actually, we don't use a spacecraft much.

Â What we really care about in this particular example is here, the t frame.

Â That's the topological frame and you can see, I've got an east access.

Â I've got a north axis and an up axis.

Â So if you look down on the up axis, you would see just like a regular map.

Â You open up, I used to say, Rand McNally but

Â who here has actually used a Rand McNally?

Â Excellent. I don't feel that old.

Â Thank you. I appreciate that.

Â But now we have, you've got your maps on your smartphones and your tablets, but

Â it's always typically shown by default, east to the right, north to the top.

Â That's the kind of frame that we're set up here and the sequence, you can see,

Â is east first, that's your x-axis as you would draw it.

Â You always tend to make that the horizontal one, your first axis.

Â The vertical is your second axis and then the third is the one out of the paper.

Â So east, north and up.

Â The inertial frame is to find that the classical or ECI, Earth-centered inertial.

Â It's just means at the center of the Earth.

Â You've got the n3 along the polar axis, n1 is defined toward vernal equinox.

Â It's a celestial thing and that kind of locks in where n2 is going to be.

Â The angle here that you have,

Â how far has your local longitude rotated from vernal equinox is called the ligama?

Â It varies with time.

Â So obviously, as the Earth rotates,

Â that longitude is going to slowly rotate around.

Â One revolution per day, more or less and that's what you need and let's see.

Â So we have gamma and the other angle we have is phi,

Â that's basically your latitude from the equator, so

Â that gets you over to the right longitude where you are right now.

Â Here in boulder and then you have to go up some amount.

Â 30 degrees, 40 degrees, 50, 60, depending on how high up you are on the globe.

Â Two angles and I want to find the DCM

Â that maps from N to T directly.

Â So let's see, next to Russell.

Â What was your name again?

Â >> Brett.

Â >> Brett.

Â 2:15

How many sequence of rotations do you think we're going to need to go from here

Â all the way up to this T frame?

Â >> There are three angles?

Â >> Which three angles?

Â >> I was just asking if there's was three.

Â >> No, you need this one,

Â that's gamma to go from vernal equinox to basically a line of nodes.

Â That's where your longitude intersects the equator.

Â >> It's just two.

Â >> So, we need two angles.

Â So you're saying with two rotation sequences,

Â you can go from inertial to the T frame.

Â So, the first rotation would be about which axis?

Â You can do about N and three xs with your- >> So, you would rotate about n3.

Â So your frame, you're rotating it over by the amount gamma.

Â Good.

Â And now about, now my one, two and three is lined up this way.

Â Now, about which axis would you rotate to get to T?

Â >> 2 by the latitude.

Â >> And do you rotate in a positive or negative sense?

Â >> Positive.

Â No, negative >> Negative,

Â you put your thumb on that direction.

Â I curl my finger, it turns out going downwards on the equator would actually be

Â a positive with the right hand rule.

Â We're going up.

Â So we're doing a 3, then a minus 2 rotation and

Â that'll get you this frame.

Â Everybody agree with that?

Â 3:41

>> Yeah, that would work.

Â >> Andrew, you don't look as convinced.

Â >> Aren't your vectors not the right way if you do that?

Â Don't you need to do one more rotation at the top?

Â >> One more?

Â Let's look at it in detail.

Â You're definitely on the right track.

Â If we do this, remember, this is one, two, three.

Â So my fingers, one, two, three.

Â I am doing as with Brett.

Â You did a tthree rotation, positive.

Â That's good.

Â Now you're doing a minus 2.

Â Now my one is pointing in which direction, east, north or up?

Â It's going to be up.

Â My two direction is actually pointing in your east and

Â my three direction is pointing in north.

Â 4:22

That's not how we defined our frame.

Â We need the first axis to be the east.

Â My first axis is pointing up, so I need additional rotations.

Â How can we get the first axis to point east?

Â >> Three by 90.

Â >> Yeah, you do the three now.

Â But if you just rotate the three axis,

Â that means you're rotating about your current up direction, 90 degrees.

Â Now, you're one axis point east, good.

Â That means things are in sequence, but I have my two axis pointing into

Â the planet and my three's pointing north, so I need one more.

Â What do I rotate now?

Â >> 1 by 90.

Â >> At the 1 plus or minus 90 plus and

Â that should give you east is first.

Â North is second, up is third.

Â So that sequence is very important, even in the current homework.

Â There's one I'm giving you that's BL, BR and BC and

Â I think we have BL as the first axis not BR.

Â And the reason is because that one angle goes to zero, they're the same frame and

Â then you don't have to flip your order over all of the sudden.

Â So when you get in these transformations, it's very tempting to do exactly

Â what Brett did and say, hey, that kind of lines up axis.

Â I have this axis, they line up.

Â But at the last step, always make sure the sequence is correct.

Â Otherwise, you have to do at maximum two extra rotations.

Â With two extra flips,

Â you can get from any sequence to any other sequence at that point.

Â Once you've lined up two axes, the third better lineup or somebody gave you

Â a left-handed frame and a right-handed frame, then you know you're tricked.

Â That would be a good prelim question.

Â I should do that on a prelim oral.

Â That would be so much fun.

Â 6:08

So, how did we do this mathematically?

Â Lets just talk through that.

Â This is the answer.

Â This is what we're looking for.

Â And is in terms of two angles as we were talking earlier, that's your latitude,

Â that's your locals at time.

Â How far the Earth has rotated?

Â So the Greenwich is typically a reference.

Â Zero longitude is Greenwich, England or just for historical reasons.

Â That is what we picked.

Â And then relative to that, what's your longitude, east or west?

Â That's what big lambda is, but these are all lambda is fixed and

Â MNG varies for time.

Â So just total angle, that's what we care about.

Â 6:40

You can look at this as a three rotation, but

Â it's the sum of the angles that we really have to track and that's gamma.

Â So, that's what we have there.

Â We're doing a three rotation, positive you plug that into our unitary DCM.

Â We're doing a single primary access rotation.

Â These things ave very handy.

Â We use them all over the place.

Â That will map me from inertial, now to this intermediate frame and

Â I'm not labeling it here.

Â But if this gets confusing to you, these intermediate frames,

Â I do recommend call it.

Â Well, that's my e-frame, an e-frame.

Â Give it a name.

Â If you get comfortable with this,

Â sometimes if all I needed is a stepping stone, I may not even label it.

Â I'll just kind of do it implied, so either way.

Â The second rotation as we're saying correctly has to be, we do the three,

Â we do the two xs, but positive would actually be going into, below the equator,

Â we need to go positive latitude to define above the equator.

Â So we give it a minus rotation about the M2 and then you plug that in,

Â cosine minus v's the same thing as cosine v.

Â It's an even function and the you have sine of minus phi is the same thing as

Â minus sin phi and that gives you all the right SIGNs that you need.

Â 7:49

The rest of it is these flippings and pretty much what we discussed, once we

Â get there, the one axis was pointing up to make the one axis point east,

Â we do that plus 30 degree rotation about our current three axis and

Â then we had to do one more to get to up to point up and north to point north and

Â that was the one rotation also plus ninety.

Â And so we can see with 90, you get very simple looking matrices which is kind of

Â nice, but you basically this allows you to rearrange this sequence, but

Â we are doing it actually here to go inertial to to the top of rapid frame.

Â We ended up having to do four sequential rotations to get to that even though it's

Â only a two degree of freedom problem, but

Â that ensures we have the right axis alignment and the ends the order.

Â The ordering sometimes is stuff that needs extra steps and

Â is often missed, that would give it to you.

Â So you do this, this is the final answer.

Â We did the three, the two, that gets us roughly with the axis kind of lined up,

Â but they're not in the right order.

Â And then we do two more rotations to kind of, those are typically always 90 or

Â minus 90, depending on what you're doing to get the sequence correct.

Â You finish up the matrix math you're done.

Â This is to me,

Â much easier than trying to do the geometric interpretation of this directly.

Â There's always some students that feel like, man,

Â I just gotta stare at it long enough.

Â Get a glass of beer, glass of wine, tea, coffee, whatever you prefer.

Â 9:14

This is boulder, water, Fiji water.

Â >> [LAUGH] >> Whatever works and stare at it.

Â But man, that's tough.

Â That's really, really tough.

Â If you can look for the sequence to go easy from this frame to this frame to this

Â frame to this frame no matter how complex the problem, you can always build

Â sequences of rotations and you now are experts on how to add these rotations.

Â