We want to talk about differentiating vectors. And immediately we need this thing called omega, body angular velocity vector. So hopefully this is very boring. Hopefully you've seen this kind of stuff before. I'm going to quick review it, put it in current context and then we'll move on. Basically, omega It's a vector, so it has a magnitude and a direction, all right? And we denote it through the Greek letter omega, usually a lower case omega, that's what you can see. And if I want to be explicit, here I'm just writing omega. If I wanted to be explicit, this angular velocity is always to angular velocity between one body and the rigid bodies represented by corner frame, a body fixed frame. So we can see how is this frame rotating relative to some other frame. And so this omega tip here it could be the angular velocity of omega. You could express it relative to N, an inertial frame, as an example. What device on a spacecraft measures omega b relative to n, the body rate is relative to the inertial frame. What's your name? >> Brian. >> Brian? >> The gyroscope? >> Yeah, the rate gyro, basically. You get these little rate gyros, they even have them in your smart phones these way. This is how we can sense orientations. You can integrate there rates with the right math and then get heading, at least differential heading. With accelerometer you can get actual, at least vertical directions. But that is what we use in spacecraft. Reg gyros, that's precisely what you measure. So that's good because that's an input. Let me put it on there, we have it. If we need it in our control, it's something we measure directly. That's really cool. It's angular velocity relative to the inertial frame. If you have to control one rate of body relative to another rate frame like an orbit frame, you want your spacecraft to line up with a rotating orbit frame, well you do not ever measure omega of the body relative to orbit. You only measure omega body of relative to N. So, let's talk through that quickly. How do we find, again, these are vectors. We want omega B relative to an orbit, and just think of a circular orbit, right? We're rotating at a constant rate, we know our reference. Once you can measure is the bond d relative to inertial, this comes from your rate gyro. What math do we have to do here to get omega B relative to O? Was it Robert? No, Theodore? You go by >> Robert. >> Sorry? >> Robert. >> Robert, okay, what math do we have to do here? I need angle velocity of b relative to o. You measure angular velocity of B relative to the inertial frame. What else do you have to do here to make these things match? Go, distance between your. >> No, distance doesn't come in. It's purely angular rate stuff. >> When you. >> So this, this quickly confuses people, which is good but it's an easy fix. Let's reverse the problem, forget the rates. We know the position of Bob. We want the position of Bob relative to O. A name with O, Otto, sorry, that's German. But still, I know. r B relative to O. And that's what you want. But what you measure is Bob relative to Nadine. In positions, how do I get a relative position? I know Bob relative to Otto. I'll pretend to be Otto. And then we have Nadine and I need Bob relative to Nadine. What math do we have to do to get the relative position between Bob and Otto? >> We need to find the position between Otto and Nadine. >> And do what? >> Add them together, or no subtract. >> Subtract, yep. In this case we subtract. People are very comfortable in positions, they know. Okay the position of this person relative to this and we know the position of that person relative to this person. I need the position of this person relative to that. I take one minus the other, you get your relative position. That's it, right. It's really that simple and that's because they're vectors, so to have these additive properties, subtraction properties, you can do this directly in vectorial form. Angular velocity is also a vector. Which means, I can do exactly the same here. If I meet the angle of velocity of this astronaut B relative to the orbit frame, but the astronaut is measuring its inertial angle of velocity, you need to have the orbit frame inertial angular velocity and subtract that from the other, then you get the relative one, right. Or you can reverse it if you wanted to have B relative to N, which we've measured B relative to O and you know O relative. You bring this over to the left-hand side. In that case, you add them, right. So get very comfortable with this because it's something you have to do a lot including in the first homeworks. If this is confusing, it's simply vector addition and subtractions over. Think of it like relative positions, but instead of relative positions, we have relative angular weights between two frames. That's it. And then you can add and subtract and nothing should confuse you, okay? And to this notation, B relative to O, so this is the angular rate that B has. If you're moving at 6 degrees per second about some axis with respect to the O frame, that's what we've got here. Very simple, so good. We got that. So, we typically express the angular velocity vector in B frame components. So, here I've written in a vectorial form. Omega as a vector is a magnitude direction, magnitude direction, magnitude direction all consistent. And then here, this is actually a matrix representation of a vector. And so I've just written omega 1 2 3. And it's implied Omega1 is about the B1 axis, Omega2 is about the B2 and Omega3 is about the B3. And I'm choosing 1,2,3 just to keep my life simple. I'm not switching threes and twos like I did earlier. That was more of an example. This is really what you do typically make it as simple as possible. So good, so the angular velocity vector what you need to remember is the angular velocity vector is the angular rotation of one frame relative to another. And it is really a vector that means it satisfies vector addition and subtraction. So if you know the rate of A relative to B And you know the rate of C relative to B, you can then add subtract them as need to get anything else. What if you have, last question, who haven't I bugged yet. Corner, what's your name? >> Kyle. >> Kyle, thank you. >> If you have omega BN, but for some reason in your math you need omega NB, just the opposite. How do those two things relate to each other? >> So you're saying you're in the same frame and you're trying to get from- >> No quoting frames. >> Okay. >> It's just a vector. All right. Magnitude time the direction. So the rotation of B relative to this inertial would be some vector. Let's say this pen pretends to be that vector, right. So, this frame is rotating at one degree per second about this axis. That's B relative to N. For some reason, I need to know how is N rotating relative to B inverting that. How do those two things relate? >> Well, I mean if you know the rotation of one axis, can't you assume that the rotation the other axis is going at the same rate because it's spinning about? >> About what? >> I don't know actually. >> You're very close. Mariele, can you help him? What's the answer? If you have omega BN. >> Reverse. >> How does it relate to NB? >> The reverse or the opposite. >> And mathematically that is? >> Negative. >> Negative, right. It's the same that doing one degree per second will, if I'm rotating at one degree per second relative to you guys about this axis. You guys are actually rotating at one degree per second about the other. Because you seem to be going, as I'm going to my left you guys seem to be going in the opposite direction. So that's there, right? Positive rotations, how do we figure out what's a positive rotation? Andrew. >> Right hand rule. >> Which is? >> Thumb goes where? >> Along the axis. >> Right, so if you have an arrow drawn, you put your thumb along that axis with your right hand. Again put an r on your exam on your hand just so you remember. So you got this and then you curl your fingers right? That is a positive rotation. That satisfies right hand rule. That's the easy quick trick, right? So good, we've got everything we need to know about omegas. Really, that's it, we've summarized them. Hopefully, you've seen this before. It's just kind of, we'll be putting it in new context.