So, at the stage, we was review last time, Andrew, without looking at your notes, what did we do? >> We went over Euler triangles. >> Okay, Kevin what are Euler angles? >> The sequence of rotations. >> Right, and that's the key. How many sequences to be used for 3D attitude? >> Three. >> Three. You might seen robotics six Euler angles like quantities, just because they have six joints in that robot manipulator. Fine, but in general, free tumbling. This is a three degree in problem, we need three angles. And we do this as a sequence. It's not that the space craft has to actually move like a robot, and doing this kind of a thing. It's just a mathematical description on how it gets to the final path. Good, so, all your angles are a sequence. So let's see, sorry. Last row, what was your name again? >> Spencer. >> Spencer, thank you. How many sets of Euler angles are there? >> What do you mean by that? Sets of rotations? >> Yeah. So let's say we have yaw, pitch, roll. What kind of a sequence of rotations was yaw, pitch, roll? So that's one set of other angles. How many other sets, theres other sequences that we can do? >> Three Not quite, 12. Why 12? Yes, so the first axis you can do one two or three. So were not doing rotations about some axis skewed with the original frame. You only have three options, either your first, your second, or your third base vector, that's how these things are typically defined. So your doing a three, a two and a one. First one you got three options, but the second one you can either, you can only do one of the other two axis, not the same axis again, otherwise you end up getting so least off. So 12 sets, these 12 sets, how do we break them down? There's groupings we have of these Euler angles. Eduardo, how do we break up the Euler angles? [INAUDIBLE] Symmetric or asymmetric. >> Symmetric and asymmetric, would 3-2-1 be a symmetric set or an asymmetric set? >> Asymmetric. >> It's an asymmetric set. So a symmetric would be 3-1-3. Good, why does it matter if it's symmetric or asymmetric? Was it David? Why does it matter if it's symmetric or asymmetric? >> I'm not sure. >> The singularities. >> That's right, it impacts. So next to David, let me just go down the row. I don't your name anymore. >> Russell. >> Russell, thank you, yes. >> The singularities are different. >> Right, now what makes an ordinary angle set go singular? Which of the three angles is a troublemaker? Second. It's always the second angle. So what's different Russell between symmetric and asymmetric sets? On the asymmetric set like this for what angle, second angle, does your description have issues? >> [INAUDIBLE] >> Plus minus 90 exactly and that's true for all the symmetrics. And if it's asymmetric like a 313 again it's the 2nd one. Now, what singularities Maurice do we have? >> For which one? >> 313 a symmetric set. >> 1. The 2nd one. >> It's the 2nd angle. But at what angles do you go singular? >> Minus 90 and 90. >> That's what asymmetric said. That's what Russell just said. >> So the other one. >> Right that's the other option. So for orbiting inclinations that's the easy way I remember this. Orbiting inclination you've got your ascending node. Inclination argument periods. It's a 313 sequence, zero inclination. You do two rotations about the three axis and you can do but you can imagine the inverse mapping is going to have all kinds of mathematical issues because of ambiguities. There's infinity of angles that sum up to be that one sum that you need. So symmetric, asymmetric, right away I can say well that's where the trouble's going to be, right. I don't even have to look at the math, I know right away. This stuff goes.