0:06

So the next topic.

Â This is kind of a big one-third of the chunk of the class is

Â Rigid Body Kinematics.

Â So I need a rigid body, he's a some a little object, it's all rigid.

Â I can throw it around, it doesn't change shape,

Â it doesn't deform at least if I don't throw it too hard.

Â So David, what does kinematics mean?

Â Yes?

Â >> Daniel.

Â >> Daniel, darn.

Â D, I was close.

Â [LAUGH] I should know.

Â Thank you.

Â >> It's the study of motion of whatever,

Â and just that motion through space.

Â >> So if I'm throwing it up, are you studying how gravity affects and

Â makes this thing fly?

Â >> No, I'm confused now.

Â >> [LAUGH] >> Not sure anymore.

Â >> Only subtleties, see the study of motion which is true.

Â But that's a really big umbrella, right?

Â What is the specific that we're doing in kinematics?

Â >> No external forces.

Â >> Yeah, no forces, no torques.

Â So what are we looking at?

Â We can cut away everything we're not looking at.

Â What remains?

Â Andrew, I asked you this last time.

Â You gave me very large question.

Â >> I thought we were done.

Â >> [LAUGH] >> It's the description of motion.

Â >> That's really the key.

Â It is the description of motion.

Â If you say study of motion, that can include its description.

Â But it also can include prediction of where you're going to be five minutes from

Â now, looking at gravity, forces, all this stuff, right?

Â So kinematics is purely about how do I describe my position.

Â If we talk translation first, right?

Â That's often easier for people to kind of visualize.

Â I know, this is my point, the p of interest.

Â Here's my coordinate frame down to the floor, that's my position.

Â Now what type of coordinates could you use to define that?

Â What was your name again?

Â >> Maurice.

Â >> Maurice, thank you.

Â >> What kind of coordinates?

Â >> Yes, just a regular position.

Â >> Sure, you could use Cartesian coordinates.

Â >> What else?

Â >> Rotational, cylindrical, Maurice's fancy pants coordinates.

Â >> Fancy pants coordinates, always popular.

Â Yep, absolutely.

Â >> [LAUGH] >> Yeah, lots of them, right?

Â But there's combinations, there's infinitesimal combinations,

Â it has an infinity of coordinates.

Â There's lots and lots of ways.

Â But people love Cartesian at the beginning,

Â because that's what they're used to, right?

Â But then as you get into more complicated systems, you've got a space craft crawling

Â on a series, which is a pretty spherical looking asteroid.

Â It's huge, right?

Â That's the one that has all the white dots recently that people have.

Â So if this is a sphere,

Â this Cartesian well now you're going to have to have these constraints.

Â That x, y, and z, all squared have to sum up to the radius of this asteroid, right?

Â That's a pain to keep track of, versus if you now switch to spherical.

Â Yeah, spherical a little bit more complicated.

Â Lots of sines and cosine, and little bit different coordinates and relationships.

Â But now all of a sudden that beam on the surface just means your radius has to be

Â constant, right?

Â And now you're acting on it.

Â Life becomes much easier, actually.

Â The control of this becomes much easier, and the same thing happens in attitude and

Â that's what we're covering right now.

Â But instead of translation with your vectors, attitude is not a vector.

Â That's one of those easy quick answer questions, you might have on an exam.

Â Is attitude of vector?

Â And if you say yes, I'm just going to blast you, so no pressure.

Â So rigid body kinematics is the description of the orientations,

Â how we get there.

Â And we're going to go through a whole series of these.

Â Today, we're going to cover the direction cosine matrix.

Â This is sometimes also called the transformation matrix or

Â the rotation matrix, I typically call it DCM for short.

Â In space, everything has to have acronyms.

Â Otherwise, it's not respected.

Â So this is called DCM, that's one of them.

Â Euler angle sets, this is hopefully something you've seen.

Â So we're going to breeze through these pretty quickly.

Â But the key elements as always, for every single set, how do I add them,

Â how they defined, where are they singular, how do I subtract them?

Â If I have relative orientations and in particular, how do I map back and

Â forth to the DCM?

Â You will find reasons why in a shortly.

Â 4:07

There's other things called principal rotation parameters, Euler parameters,

Â classical Rodrigues Parameters, Modified Rodrigues Parameters.

Â We'll even talk about stereographic orientation parameters.

Â This is a whole series and there's lots more that can be defined.

Â There's still papers being published to this day.

Â On different kind of attitude descriptions, and

Â how we go back and forth.

Â And so as we go through them at the beginning is a little bit more detailed.

Â We're covering a lot of, what exactly does this stuff mean?

Â How do these things relate?

Â How do you get to them?

Â As we get to the lower part of these coordinates,

Â we'll start to accelerate a little bit,

Â because it's kind of the same concepts over and over.

Â But now we're using this definition, then this definition.

Â And what I expect for you to have by the end of this whole thing is a really

Â intimate understanding of how to describe orientations.

Â And if somebody gives you coordinates that you've never seen before,

Â you should not flinch.

Â And go, you know what?

Â This is what I have to find, and now I'm ready go.

Â And there's some clear ways to deal with it.

Â This is a topic that's very, very popular with people, especially once they leave

Â this class and university and go work at JPL or or who knows where.

Â because 3D orientations, to this day, still confuses a lot.

Â And how you define it, and what does it mean here, or

Â the notation isn't always very clear.

Â So we're going to try to do this in a very nice, clean, easy to use, vigorous way,

Â that's one of them.

Â So many of you will have seen sets of this part of you, but

Â none of you probably has seen all of this stuff.

Â Especially when we get to some of these things that will be a lot of of head

Â scratching and that's fun, right?

Â You're here to expand your envelope, so attitude coordinate sets.

Â 5:35

One of the definitions is basically, if we're talking about the orientation of

Â a rigid body, we're here trying in this spacecraft.

Â And if attached is b frame, this is a body fixed frame.

Â If you're talking mathematically about the evolution of

Â the orientation of this frame.

Â Or you're talking about the evolution of the orientation of this body,

Â there equivalent, right?

Â Because b is a body fixed one.

Â So if you know, this frame has pitched up 90 degrees,

Â the body must have pitched up 90 degrees.

Â Now how many ways can you attach a frame to a body?

Â Sorry, what was your name?

Â Yes.

Â >> Spencer.

Â >> Spencer, thank you.

Â >> As many as you want.

Â >> Yeah, as in infinity, right?

Â We typically line them up with symmetry.

Â If you're an airplane, there's the roll axis, the pitch, the yaw and

Â you'll see the different definitions for this stuff, and that's fine.

Â But there is really an infinity of ways cause even you look at the space station

Â you might be a flight instead of access.

Â If you're in a docking port, that's maybe a docking port access where you really

Â worry about all the orientations relative to docking port.

Â If your star tracker does probably a frame align the star track,

Â if IMU is very likely there is a frame relied with the IMU.

Â So even on a single rigid body in actual flight software, you might have six,

Â eight, ten different frames to find.

Â Now if it's rigid, all these orientations are fixed relative to each other.

Â So once I know one, this is my primary frame and

Â the start tracker is rolled 90 degrees down that's where it's pointing.

Â Well no matter what the body does, I know it's plus a 90 degree rotation and

Â then where does that go, right?

Â And as a plus not in a sense of adding vectors,

Â you will see the math we have to do to add rotations and subtract rotations.

Â So good, so in this section, you won't see me draw a lot of space crafts.

Â I'm not very good at that, I'm reasonable actually.

Â I shouldn't been on by myself.

Â But the frames, you [INAUDIBLE] always draw frames, that's it.

Â That represents whatever rigid body we have going.

Â There's infinite number of these, just as with translation we're talking about,

Â there's lots of different combinations.

Â There literally is infinite numbers.

Â There's whole families that have actually published on,

Â where there's parameters that you can infinitely test vary.

Â And I can create parameters for

Â you that will go singular at any desired orientation.

Â Anybody knows attitude set that goes singular at some point?

Â You guys are way too shy.

Â >> Euler angles.

Â >> Thank you, you took 3,200.

Â You better know that, right?

Â Euler angles.

Â So Kaylee, let's say a three, two, one, yaw, pitch roll, right?

Â The classic yaw, pitch roll.

Â Which angle goes singular?

Â >> It's pitch, 90 degrees.

Â >> 90 degrees, right.

Â It's at 90 degrees is a mathematical issue, that we have to deal with.

Â Some of the time is issue is just a zero over zero.

Â There's an ambiguity, there's an infinity of ways this coordinates could be defined

Â at this orientation.

Â That's one way the single area just can manifest.

Â The other way is your coordinates literally blow up to infinity.

Â They just become infinitely large in that set, and

Â we'll see both sets that do this kind of stuff.

Â If they go off to infinity, that's actually nice from a control perspective.

Â 9:18

So a good choice of coordinates,

Â we're talking about moving on an asteroid, right?

Â If you're moving on a spherical balloon, a spherical shape, and all you're doing is

Â using Cartesian coordinates, you deserve what you get, to be simple.

Â All right, there's much better coordinates that would make your life so much easier.

Â You just have to put a little effort into the kinematics.

Â How do I want to describe my motion?

Â This is where laziness is good.

Â So you want to make this as simple there as possible.

Â But you have to understand often the simplicity comes at a dealing with

Â rotating systems and

Â rotating frames, which you're now practicing with the transport theorem.

Â Once you leak that part, you are ready to go and this will be much, much easier.

Â A bad choice as we said, could really have a lot of issues.

Â It could make your life more complicated to control,

Â it could make your life more complicated in your dynamic stability analysis.

Â But it could also lead to singularities.

Â If we would use, as Kaylee was saying, the on pitch rule for general rotations.

Â Nothing says a fighter jet can't pitch up 90 degrees.

Â The F16 has more thrust than weight.

Â It takes off and goes off like a rocket.

Â Doesn't even need wings, really.

Â Except for landing, detail, but there's parachutes.

Â You're good.

Â >> [LAUGH] >> So if you're going off and doing that,

Â the plane can physically pitch up.

Â You've chosen a description that happens to go singular.

Â But physically, nothing prevents the plane from going up.

Â Why did you pick coordinates that actually would prevent the control from allowing

Â the pilot to do such a maneuver.

Â So same thing happens in space craft,

Â you see a lot of people that have historically used yaw, pitch roll.

Â There was a big government space sponsor, not to be named and embarrassed.

Â But you know who I mean, starts with N.

Â Who always used to require 20 years ago, all the controls,

Â everything has to be done, not the controls.

Â But all the analysis, everything has to be done in Euler angles,

Â because they're easy to understand.

Â I'll show you why that's not exactly the case, especially not for large rotations.

Â Because now you're introducing singularities and

Â you control your estimation.

Â All is handicapped by your choice of kinematics and

Â there's other sets that are much more applicable for large rotation.

Â Every single set of coordinates we use has an application.

Â And you will find cases where wow, I'm glad I know Euler angles.

Â I'm glad I know DCMs, I'm glad I know the quaternions.

Â But you need to be familiar with, this is like your toolbox, right?

Â You don't just want one wrench one size in there,

Â you want a whole set of tools to understand whatever the problem is.

Â I have the right tool that's going to do the job effectively.

Â That's essentially what we're learning in this section, and

Â it's a good chunk of the class.

Â Everything kind of builds on this.

Â [COUGH] Fundamental differences and orientations, so

Â back to the translation example.

Â 11:48

If this is my position, and then you have something else moving relative to it.

Â So we got two objects, they can be very close, they can be very far.

Â How far apart could they be?

Â Is it Kevin?

Â >> Yeah.

Â >> Yes, how far apart can you be translationally?

Â And we're dealing with Euclidean space.

Â We're not going into Einstein's and curvatures and stuff.

Â >> And it can be any distance?

Â >> Yeah, any means infinite, right?

Â So your tracking errors can actually become infinitely large.

Â So for all practical purpose, that's good.

Â Well but it also means, if you're doing feedback control, you're doing gain study.

Â If you're doing a simple feedback, k times tracking error, and

Â tracking error goes to infinity as you go far far away.

Â You're hitting it with a huge control effort which is very

Â likely going to saturate it.

Â And now maybe your stability analysis may breaks down or other issues might come in.

Â And we'll see saturation when we get to the control side of the class.

Â Attitudes though are a very different beast.

Â So with attitudes, if you're looking at this, just with the one attitude.

Â If this is where I'm supposed to be pointing then for

Â some reason not paying attention, I'm talking to this wall.

Â 12:58

My attitude area is about 90 degrees.

Â If I'm really tired, and I'm talking like this all lecture long,

Â you should probably throw tomatoes at me.

Â But my attitude there is 180 degrees.

Â If I keep on wandering and start talking this way,

Â my attitude there is 270 degrees.

Â Is 270 worse than 180?

Â Sorry, what was your name?

Â >> Rick.

Â >> Rick.

Â Brett, sorry.

Â >> Brett, sorry.

Â >> What do you mean by worse?

Â >> I'm making the numbers bigger.

Â This was zero, this was good.

Â 90, 180, 270, it sounds like 270's worse than 180.

Â >> Better.

Â >> It's actually better, right.

Â So the fundamental thing that's built into attitudes is there is a finite set.

Â Mathematically, it's called the SO3 groups.

Â If you read papers in attitude stuff and see SO with the 3,

Â it's a mathematical description of this group.

Â That's what we belong to, not the attitude coordinates.

Â They're not part of a vector group, where you can do vector addition,

Â subtractions and so forth.

Â SO3, so there's a finite limit.

Â You can rotate, rotate, make it bad, get as bad as it gets, and that's 180 degrees.

Â You would never have infinite angle, because at some point,

Â you're going to revolve again about the orientation you want to have.

Â This is really important in control.

Â You will see this when we try to do detone in control.

Â Some descriptions automatically take advantage of this, others not so

Â much, right?

Â And so depending on what you're doing, they have different benefits and

Â drawbacks.

Â This is very practical, so I can tell you, teaching skydiving, right?

Â Or hanging out, outside the plane.

Â It's a high winged plane, we had a Cessna 182, little strut.

Â The student stands out there, this is an AFF program, solid free fall.

Â One jump master holding on the left, one jump master holding on the right, and

Â then we are supposed to leave together.

Â The student's supposed to up, down, and step gently off into the wind arch.

Â Put hips forward, arms back, legs back, so

Â you kind of fly like a butterfly, t that's a theory.

Â The reality, lot of profanity, [NOISE] pushing, shoving, kicking and off you go.

Â And there's at this one video I got in California that was really cool.

Â I did this jump at the jump master, we went and

Â the student really shoved off hard.

Â And we're both trying to stabilize it, but there was so

Â much momentum put into the system what was basically going up side down and

Â look over smiling at each other, going what the hell?

Â And we just basically complete the tumble, flatten out, fly on the rest,

Â student had no clue.

Â This was their first jump, you get blackouts and everything.

Â But that's basically it, instead of trying to unwind it and fight it all the way,

Â at some point you're going kind of attitude perspective.

Â It's easier to just hlip it all the way around and recover here again,

Â if you're unconstrained.

Â If you're a mechanical system that has wires attached and you can only rotate so

Â far before you rip out wires, well then it really matters.

Â You really want to make sure you don't just cut that thing loose.

Â But in space, we're typically tumbling freely, there's no constraints.

Â So again, these tumbling things,

Â you will see something throughout the class we keep talking about.

Â Why did we describes one way versus another?

Â Which one helps us into control, which helps us in the dynamic prediction?

Â Attitude errors can only get up to 180 degrees.

Â 4 Truths, need a minimum of three coordinates.

Â We live in a three dimensional world, right?

Â So your position has three coordinates thing, but

Â also your attitude has three degrees of freedom.

Â So your orientation needs a minimum of three coordinates to

Â define any general attitude, okay?

Â That's easy, but the minimum sets of three coordinates.

Â We'll have at least one geometric

Â orientation where the description becomes singular.

Â And as we just mentioned, singular might mean ambiguities,

Â because zero over zero in the math.

Â Because this angle could be an infinity of angles.

Â And you will see different cases of that, or

Â it means your coordinates literally go off to infinity.

Â You've got something finite over zero at that orientation, and now big,

Â big numbers.

Â So both ways you have issues, those are called singularities.

Â So at or near the singularity,

Â the kinematic differential equations are also singular.

Â We're going to develop this equation, this is basically the equation that relates

Â this omega that we've developed, the angular velocity vector omega.

Â That's what we measured in [INAUDIBLE].

Â That's what we use to actually develop and

Â derive our equations of motion in the future.

Â But when we integrate and have yaw angle,

Â pitch angle or quaternion rates or DCM rates.

Â We have attitude coordinate rates that has to relate to these omega,

Â that's our differential kinematic equation.

Â We can see it over and over and over and over again.

Â But that's the equation that would go zero over zero or

Â something over zero which makes this whole thing blow up.

Â Now this singularities or these zero issues can be

Â avoided by moving to more than three coordinates.

Â So you need three just to do generally tumbling objects and there's coordinate

Â sets that can avoid singularities at the cost of additional coordinates.

Â Anybody here heard of coordinates that's non-singular?

Â >> Quaternions.

Â >> Quaternions, Eulers parameters is one of them.

Â And there's different sets we'll see, actually.

Â There's several sets that come up here that'll be non-singular,

Â but it's always four or more.

Â So let me invert that, several of you have heard this before.

Â If I have a set with four coordinates, am I guaranteed, it's non-singular?

Â [COUGH] Luis?

Â >> Yeah.

Â >> No, because you could just have a redundant set that doesn't-

Â >> Yeah.

Â >> Add any.

Â >> So just having four coordinates doesn't guarantee that you're non-singular.

Â But it allows the option of having a smart set and there's lots of different sets.

Â And chapter three, we'll be going through most of them,

Â not quite all of them because there's a bunch of them that do that.

Â So we have to always go to more than four.

Â