Hi and welcome to module three of Three Dimensional Dynamics. Last time we defined the angular velocity in three dimensions. And today's learning outcome is to define the properties of that angular velocity for three dimensional motion. So here's where we left off last time. We had a three dimensional motion of a robotic body. Here's the generic theory with the derivative formula that we had derived. And here's the expression for the angular velocity including now i, j, and k components. So this angular velocity has several properties. First of all, it makes sense that we have these two frames. The angular velocity depends intimately on the way the frame B changes its orientation with respect to frame F. That goes without saying, I guess. We also can prove that the angular velocity is unique, which means there is only one angular velocity at B with respect to F. Now as far as the proof is concerned, you can find it in several textbooks. The textbook that I'm using in this course is a textbook, Engineering Dynamics by two of my colleagues at Georgia Tech, Professors Dave McGill, and Wilton King who are both professor emeritus from Georgia Tech. They were very kind in allowing me to use several examples and figures from their texts in this course and it's very much appreciated. So you can find that proof in their book. I'll go through one proof together with you, but some of them I'm gonna leave that proof up for you to look it up on your own. Another property is that if frame F and frame B maintain a constant orientation, then the angular velocity of B with respect to F is 0. And again that should make intuitive sense. Here is another one that should make intuitive sense but can also be formally proved. The angular velocity of frame B with respect to frame F is negative the angular velocity of frame F with respect to B. So if you get an angular velocity of B with respect to F, if you look at it from B back towards F they're just gonna be negative of each other. And again, that should make intuitive sense. So let's look at one more important property of angular velocity, and that's called the Addition Theorem. In this situation, I've now introduced another reference frame, so I have my frame F attached to the base of the robot's head. I've got a frame B here attached to this portion of the robotic arm, and then I have another frame C attached to this portion of the robotic arm. And the addition theorem is going to allow us to relate the angular velocities of those frames with respect to each other. And so to do that, we're going to use the derivative formula, which I show here, and so this is the derivative formula for expressing a derivative of the vector A in the C frame with respect to the F frame. Then I can also write the derivative, use the derivative formula to write the derivative of A expressed in the C frame with respect to the B frame. And finally I can write the derivative formula for A expressed in the B frame with respect to the F frame. And then I can now, with those three expressions, I can add 2 and 3. And when I do that, AB is on both sides so it's gonna cancel. And, I end up with the derivative of A in the F frame, is equal to the derivative of A in the C frame, plus now we have two omegas. Omega C with respect to B, and omega B with respect to F crossed with the vector A. And then I can now compare equation 1 with equation 4, and I see we have the derivative opf A with respect to the F frame on the left-hand side, the derivative of A with respect to C on the right-hand side, and then this omega has to be equal to this omega, because it's crossed with A. And so this is what we call The Addition Theorem. We see that the angular velocity of the frame C with respect to the frame F is equal to the angular velocity of the frame C with respect to B, plus the angular velocity of the frame B with respect to F. And so you add the angular velocities of C with respect B, and B with respect to F to get the angular velocity of C with respect to F, and so we say that's been proved. Latin, we say Q.E.D., which translates to which had to be demonstrated, and important note is that this may be extended to any number of frames. I did it for these 3 frames but you can have several more frames as you get more complex three dimensional motion. And so that's it for this lesson and we'll see you at the next module.