So the next topic. This is kind of a big one-third of the chunk of the class is Rigid Body Kinematics. So I need a rigid body, he's a some a little object, it's all rigid. I can throw it around, it doesn't change shape, it doesn't deform at least if I don't throw it too hard. So David, what does kinematics mean? Yes? >> Daniel. >> Daniel, darn. D, I was close. [LAUGH] I should know. Thank you. >> It's the study of motion of whatever, and just that motion through space. >> So if I'm throwing it up, are you studying how gravity affects and makes this thing fly? >> No, I'm confused now. >> [LAUGH] >> Not sure anymore. >> Only subtleties, see the study of motion which is true. But that's a really big umbrella, right? What is the specific that we're doing in kinematics? >> No external forces. >> Yeah, no forces, no torques. So what are we looking at? We can cut away everything we're not looking at. What remains? Andrew, I asked you this last time. You gave me very large question. >> I thought we were done. >> [LAUGH] >> It's the description of motion. >> That's really the key. It is the description of motion. If you say study of motion, that can include its description. But it also can include prediction of where you're going to be five minutes from now, looking at gravity, forces, all this stuff, right? So kinematics is purely about how do I describe my position. If we talk translation first, right? That's often easier for people to kind of visualize. I know, this is my point, the p of interest. Here's my coordinate frame down to the floor, that's my position. Now what type of coordinates could you use to define that? What was your name again? >> Maurice. >> Maurice, thank you. >> What kind of coordinates? >> Yes, just a regular position. >> Sure, you could use Cartesian coordinates. >> What else? >> Rotational, cylindrical, Maurice's fancy pants coordinates. >> Fancy pants coordinates, always popular. Yep, absolutely. >> [LAUGH] >> Yeah, lots of them, right? But there's combinations, there's infinitesimal combinations, it has an infinity of coordinates. There's lots and lots of ways. But people love Cartesian at the beginning, because that's what they're used to, right? But then as you get into more complicated systems, you've got a space craft crawling on a series, which is a pretty spherical looking asteroid. It's huge, right? That's the one that has all the white dots recently that people have. So if this is a sphere, this Cartesian well now you're going to have to have these constraints. That x, y, and z, all squared have to sum up to the radius of this asteroid, right? That's a pain to keep track of, versus if you now switch to spherical. Yeah, spherical a little bit more complicated. Lots of sines and cosine, and little bit different coordinates and relationships. But now all of a sudden that beam on the surface just means your radius has to be constant, right? And now you're acting on it. Life becomes much easier, actually. The control of this becomes much easier, and the same thing happens in attitude and that's what we're covering right now. But instead of translation with your vectors, attitude is not a vector. That's one of those easy quick answer questions, you might have on an exam. Is attitude of vector? And if you say yes, I'm just going to blast you, so no pressure. So rigid body kinematics is the description of the orientations, how we get there. And we're going to go through a whole series of these. Today, we're going to cover the direction cosine matrix. This is sometimes also called the transformation matrix or the rotation matrix, I typically call it DCM for short. In space, everything has to have acronyms. Otherwise, it's not respected. So this is called DCM, that's one of them. Euler angle sets, this is hopefully something you've seen. So we're going to breeze through these pretty quickly. But the key elements as always, for every single set, how do I add them, how they defined, where are they singular, how do I subtract them? If I have relative orientations and in particular, how do I map back and forth to the DCM? You will find reasons why in a shortly. There's other things called principal rotation parameters, Euler parameters, classical Rodrigues Parameters, Modified Rodrigues Parameters. We'll even talk about stereographic orientation parameters. This is a whole series and there's lots more that can be defined. There's still papers being published to this day. On different kind of attitude descriptions, and how we go back and forth. And so as we go through them at the beginning is a little bit more detailed. We're covering a lot of, what exactly does this stuff mean? How do these things relate? How do you get to them? As we get to the lower part of these coordinates, we'll start to accelerate a little bit, because it's kind of the same concepts over and over. But now we're using this definition, then this definition. And what I expect for you to have by the end of this whole thing is a really intimate understanding of how to describe orientations. And if somebody gives you coordinates that you've never seen before, you should not flinch. And go, you know what? This is what I have to find, and now I'm ready go. And there's some clear ways to deal with it. This is a topic that's very, very popular with people, especially once they leave this class and university and go work at JPL or or who knows where. because 3D orientations, to this day, still confuses a lot. And how you define it, and what does it mean here, or the notation isn't always very clear. So we're going to try to do this in a very nice, clean, easy to use, vigorous way, that's one of them. So many of you will have seen sets of this part of you, but none of you probably has seen all of this stuff. Especially when we get to some of these things that will be a lot of of head scratching and that's fun, right? You're here to expand your envelope, so attitude coordinate sets. One of the definitions is basically, if we're talking about the orientation of a rigid body, we're here trying in this spacecraft. And if attached is b frame, this is a body fixed frame. If you're talking mathematically about the evolution of the orientation of this frame. Or you're talking about the evolution of the orientation of this body, there equivalent, right? Because b is a body fixed one. So if you know, this frame has pitched up 90 degrees, the body must have pitched up 90 degrees. Now how many ways can you attach a frame to a body? Sorry, what was your name? Yes. >> Spencer. >> Spencer, thank you. >> As many as you want. >> Yeah, as in infinity, right? We typically line them up with symmetry. If you're an airplane, there's the roll axis, the pitch, the yaw and you'll see the different definitions for this stuff, and that's fine. But there is really an infinity of ways cause even you look at the space station you might be a flight instead of access. If you're in a docking port, that's maybe a docking port access where you really worry about all the orientations relative to docking port. If your star tracker does probably a frame align the star track, if IMU is very likely there is a frame relied with the IMU. So even on a single rigid body in actual flight software, you might have six, eight, ten different frames to find. Now if it's rigid, all these orientations are fixed relative to each other. So once I know one, this is my primary frame and the start tracker is rolled 90 degrees down that's where it's pointing. Well no matter what the body does, I know it's plus a 90 degree rotation and then where does that go, right? And as a plus not in a sense of adding vectors, you will see the math we have to do to add rotations and subtract rotations. So good, so in this section, you won't see me draw a lot of space crafts. I'm not very good at that, I'm reasonable actually. I shouldn't been on by myself. But the frames, you [INAUDIBLE] always draw frames, that's it. That represents whatever rigid body we have going. There's infinite number of these, just as with translation we're talking about, there's lots of different combinations. There literally is infinite numbers. There's whole families that have actually published on, where there's parameters that you can infinitely test vary. And I can create parameters for you that will go singular at any desired orientation. Anybody knows attitude set that goes singular at some point? You guys are way too shy. >> Euler angles. >> Thank you, you took 3,200. You better know that, right? Euler angles. So Kaylee, let's say a three, two, one, yaw, pitch roll, right? The classic yaw, pitch roll. Which angle goes singular? >> It's pitch, 90 degrees. >> 90 degrees, right. It's at 90 degrees is a mathematical issue, that we have to deal with. Some of the time is issue is just a zero over zero. There's an ambiguity, there's an infinity of ways this coordinates could be defined at this orientation. That's one way the single area just can manifest. The other way is your coordinates literally blow up to infinity. They just become infinitely large in that set, and we'll see both sets that do this kind of stuff. If they go off to infinity, that's actually nice from a control perspective. Because if you're feeding back on attitude, you do a simple linear gain minus k times your attitude error. And if you want to make sure your solar panels haven't rotated more than 90 degrees. If you have a set of altitude corners whose values goes to infinity at 90 degrees, you can become infinitely aggressive to restore your altitude back at the sun, right? Close to the sun you're kind of weaker, but then as you go up it's like a cubic or a highly non linear spring. And I can design coordinates for you that will go to infinity at any angle, or about any particular axes. That's kind of the latter part of the stuff. This is really where some head scratching comes in. Wait a minute, how did we do this? But it's really slick and cool, and it's plain fun. Nothing wrong with having fun. So a good choice of coordinates, we're talking about moving on an asteroid, right? If you're moving on a spherical balloon, a spherical shape, and all you're doing is using Cartesian coordinates, you deserve what you get, to be simple. All right, there's much better coordinates that would make your life so much easier. You just have to put a little effort into the kinematics. How do I want to describe my motion? This is where laziness is good. So you want to make this as simple there as possible. But you have to understand often the simplicity comes at a dealing with rotating systems and rotating frames, which you're now practicing with the transport theorem. Once you leak that part, you are ready to go and this will be much, much easier. A bad choice as we said, could really have a lot of issues. It could make your life more complicated to control, it could make your life more complicated in your dynamic stability analysis. But it could also lead to singularities. If we would use, as Kaylee was saying, the on pitch rule for general rotations. Nothing says a fighter jet can't pitch up 90 degrees. The F16 has more thrust than weight. It takes off and goes off like a rocket. Doesn't even need wings, really. Except for landing, detail, but there's parachutes. You're good. >> [LAUGH] >> So if you're going off and doing that, the plane can physically pitch up. You've chosen a description that happens to go singular. But physically, nothing prevents the plane from going up. Why did you pick coordinates that actually would prevent the control from allowing the pilot to do such a maneuver. So same thing happens in space craft, you see a lot of people that have historically used yaw, pitch roll. There was a big government space sponsor, not to be named and embarrassed. But you know who I mean, starts with N. Who always used to require 20 years ago, all the controls, everything has to be done, not the controls. But all the analysis, everything has to be done in Euler angles, because they're easy to understand. I'll show you why that's not exactly the case, especially not for large rotations. Because now you're introducing singularities and you control your estimation. All is handicapped by your choice of kinematics and there's other sets that are much more applicable for large rotation. Every single set of coordinates we use has an application. And you will find cases where wow, I'm glad I know Euler angles. I'm glad I know DCMs, I'm glad I know the quaternions. But you need to be familiar with, this is like your toolbox, right? You don't just want one wrench one size in there, you want a whole set of tools to understand whatever the problem is. I have the right tool that's going to do the job effectively. That's essentially what we're learning in this section, and it's a good chunk of the class. Everything kind of builds on this. [COUGH] Fundamental differences and orientations, so back to the translation example. If this is my position, and then you have something else moving relative to it. So we got two objects, they can be very close, they can be very far. How far apart could they be? Is it Kevin? >> Yeah. >> Yes, how far apart can you be translationally? And we're dealing with Euclidean space. We're not going into Einstein's and curvatures and stuff. >> And it can be any distance? >> Yeah, any means infinite, right? So your tracking errors can actually become infinitely large. So for all practical purpose, that's good. Well but it also means, if you're doing feedback control, you're doing gain study. If you're doing a simple feedback, k times tracking error, and tracking error goes to infinity as you go far far away. You're hitting it with a huge control effort which is very likely going to saturate it. And now maybe your stability analysis may breaks down or other issues might come in. And we'll see saturation when we get to the control side of the class. Attitudes though are a very different beast. So with attitudes, if you're looking at this, just with the one attitude. If this is where I'm supposed to be pointing then for some reason not paying attention, I'm talking to this wall. My attitude area is about 90 degrees. If I'm really tired, and I'm talking like this all lecture long, you should probably throw tomatoes at me. But my attitude there is 180 degrees. If I keep on wandering and start talking this way, my attitude there is 270 degrees. Is 270 worse than 180? Sorry, what was your name? >> Rick. >> Rick. Brett, sorry. >> Brett, sorry. >> What do you mean by worse? >> I'm making the numbers bigger. This was zero, this was good. 90, 180, 270, it sounds like 270's worse than 180. >> Better. >> It's actually better, right. So the fundamental thing that's built into attitudes is there is a finite set. Mathematically, it's called the SO3 groups. If you read papers in attitude stuff and see SO with the 3, it's a mathematical description of this group. That's what we belong to, not the attitude coordinates. They're not part of a vector group, where you can do vector addition, subtractions and so forth. SO3, so there's a finite limit. You can rotate, rotate, make it bad, get as bad as it gets, and that's 180 degrees. You would never have infinite angle, because at some point, you're going to revolve again about the orientation you want to have. This is really important in control. You will see this when we try to do detone in control. Some descriptions automatically take advantage of this, others not so much, right? And so depending on what you're doing, they have different benefits and drawbacks. This is very practical, so I can tell you, teaching skydiving, right? Or hanging out, outside the plane. It's a high winged plane, we had a Cessna 182, little strut. The student stands out there, this is an AFF program, solid free fall. One jump master holding on the left, one jump master holding on the right, and then we are supposed to leave together. The student's supposed to up, down, and step gently off into the wind arch. Put hips forward, arms back, legs back, so you kind of fly like a butterfly, t that's a theory. The reality, lot of profanity, [NOISE] pushing, shoving, kicking and off you go. And there's at this one video I got in California that was really cool. I did this jump at the jump master, we went and the student really shoved off hard. And we're both trying to stabilize it, but there was so much momentum put into the system what was basically going up side down and look over smiling at each other, going what the hell? And we just basically complete the tumble, flatten out, fly on the rest, student had no clue. This was their first jump, you get blackouts and everything. But that's basically it, instead of trying to unwind it and fight it all the way, at some point you're going kind of attitude perspective. It's easier to just hlip it all the way around and recover here again, if you're unconstrained. If you're a mechanical system that has wires attached and you can only rotate so far before you rip out wires, well then it really matters. You really want to make sure you don't just cut that thing loose. But in space, we're typically tumbling freely, there's no constraints. So again, these tumbling things, you will see something throughout the class we keep talking about. Why did we describes one way versus another? Which one helps us into control, which helps us in the dynamic prediction? Attitude errors can only get up to 180 degrees. 4 Truths, need a minimum of three coordinates. We live in a three dimensional world, right? So your position has three coordinates thing, but also your attitude has three degrees of freedom. So your orientation needs a minimum of three coordinates to define any general attitude, okay? That's easy, but the minimum sets of three coordinates. We'll have at least one geometric orientation where the description becomes singular. And as we just mentioned, singular might mean ambiguities, because zero over zero in the math. Because this angle could be an infinity of angles. And you will see different cases of that, or it means your coordinates literally go off to infinity. You've got something finite over zero at that orientation, and now big, big numbers. So both ways you have issues, those are called singularities. So at or near the singularity, the kinematic differential equations are also singular. We're going to develop this equation, this is basically the equation that relates this omega that we've developed, the angular velocity vector omega. That's what we measured in [INAUDIBLE]. That's what we use to actually develop and derive our equations of motion in the future. But when we integrate and have yaw angle, pitch angle or quaternion rates or DCM rates. We have attitude coordinate rates that has to relate to these omega, that's our differential kinematic equation. We can see it over and over and over and over again. But that's the equation that would go zero over zero or something over zero which makes this whole thing blow up. Now this singularities or these zero issues can be avoided by moving to more than three coordinates. So you need three just to do generally tumbling objects and there's coordinate sets that can avoid singularities at the cost of additional coordinates. Anybody here heard of coordinates that's non-singular? >> Quaternions. >> Quaternions, Eulers parameters is one of them. And there's different sets we'll see, actually. There's several sets that come up here that'll be non-singular, but it's always four or more. So let me invert that, several of you have heard this before. If I have a set with four coordinates, am I guaranteed, it's non-singular? [COUGH] Luis? >> Yeah. >> No, because you could just have a redundant set that doesn't- >> Yeah. >> Add any. >> So just having four coordinates doesn't guarantee that you're non-singular. But it allows the option of having a smart set and there's lots of different sets. And chapter three, we'll be going through most of them, not quite all of them because there's a bunch of them that do that. So we have to always go to more than four.
