The movement of bodies in space (like spacecraft, satellites, and space stations) must be predicted and controlled with precision in order to ensure safety and efficacy. Kinematics is a field that develops descriptions and predictions of the motion of these bodies in 3D space. This course in Kinematics covers four major topic areas: an introduction to particle kinematics, a deep dive into rigid body kinematics in two parts (starting with classic descriptions of motion using the directional cosine matrix and Euler angles, and concluding with a review of modern descriptors like quaternions and Classical and Modified Rodrigues parameters). The course ends with a look at static attitude determination, using modern algorithms to predict and execute relative orientations of bodies in space. After this course, you will be able to... * Differentiate a vector as seen by another rotating frame and derive frame dependent velocity and acceleration vectors * Apply the Transport Theorem to solve kinematic particle problems and translate between various sets of attitude descriptions * Add and subtract relative attitude descriptions and integrate those descriptions numerically to predict orientations over time * Derive the fundamental attitude coordinate properties of rigid bodies and determine attitude from a series of heading measurements
6: Euler Angle Definition
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来自 University of Colorado Boulder 的课程
Kinematics: Describing the Motions of Spacecraft
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从本节课中
Rigid Body Kinematics I
This module provides an overview of orientation descriptions of rigid bodies. The 3D heading is here described using either the direction cosine matrix (DCM) or the Euler angle sets. For each set the fundamental attitude addition and subtracts are discussed, as well as the differential kinematic equation which relates coordinate rates to the body angular velocity vector.
与讲师见面
Hanspeter SchaubAlfred T. and Betty E. Look Professor of Engineering
Department of Aerospace Engineering Sciences
Okay, moving on.
Euler angles.
Everybody thinks they know Euler angles, they're like,
[SOUND] I've seen that before.
Great.
So tell me, you look perky back there, back row.
Okay, grey shirt, what was your name?
Brief.
>> Maurice.
>> Me or him? >> No, no, back.
>> Right here.
Thank you. >> What are Euler angles?
>> [SOUND] I was one of the people that did not say I have seen those before.
No.
>> [LAUGH] >> My understanding is, I mean,
what comes to mind is rotation about more or less, a [INAUDIBLE] coordinate frame.
>> Okay. >> Theta.
>> You've definitely heard of them before.
Yes. Good.
Give me a set of Euler angles that you're very familiar with.
>> Pitch roll.
>> Yeah, pitch roll, that's the classic one.
If you're doing orbits, can someone give me an orbit set of Euler angles?
>> Right ascension inclination and argument?
>> Yeah.
If you have orbits, they don't typically call them Euler angles.
They just call them ephemeris.
These are orbit angles and so forth.
But in fact they are Euler angles.
And instead of doing ephemeri in, those are three one three sets actually,
molar angles, you'll see papers where people use quaternions
to define the orbit plane, or MRPs to define orbit planes.
And there's whole other ways of defining these orientations of the plane.
So you can see all these worlds kind of cross couple like that.
Good.
But basically, as we say, something like yaw, pitch and roll.
And again, NASA was the big culprit there for awhile,
claiming that they're easy to visualize.
Who's had yaw pitch roll before?
Okay.
Perfect.
Evan, easy to visualize?
>> Depends on what the object is, I think, but yes.
>> If I give you 3 degrees yaw, 2 degrees pitch, 1 degree role,
could you show me with your fingers what that attitude is?
>> Yeah, so I already forgot the numbers you said.
>> Three, two, one.
3 degrees yaw, 2 degrees pitch, 1 degree roll.
>> Yes so, yaw, and then pitch and- >> Okay.
So you went through the sequence of motions, that's good.
because that is related to what you were talking about.
Euler angles are actually a sequential rotation sequence.
So this is the mathematics, we are adding.
You're doing, basically your first ration about one of these orthogonal base vectors
that Maurice was talking about.
So you have your right hand coordinate frame, B and N are equivalent, and
you're saying, I'm doing my 3 degrees yaw, then I'm doing my 2 degrees pitch.
And 1 degree rolled and to think about it for a second.
So if you have your thumps up like this, pitch is actually down,
you put your thumb along that axis curl your fingers in that case it's down.
That's why often shifts to find it differently so pitch on a ship is up.
So just be careful how those frames are defined on how we do that or
on an aircraft, we define it differently.
Where is your yaw access on an aircraft?
Down actually and where's your roll access on an aircraft?
>> Right along the.
>> Yeah, along the forefront, that's your pitch axis that you're thinking of.
So the one axis go forward,
the two axis goes to the right wing, three axis goes down.
That's why a positive yaw is to the right, a positive pitch is actually pitching up.
But a positive roll means you're tilting to the right.
That's how they define the frame.
So I'm using these numbers, it's a sequential rotation sequence.
Yes?
>> Just say positive pitches down.
>> That's how you define your thing.
If I have it this way, positive pitches down.
If I define my frame this way, then a positive pitch is up,
because one has to pitch to the left, one has to pitch to the right.
So be careful how that's actually defined.
So what we want to go through here, everybody's kind of somewhat seen this.
I'm not going to spend lots of details.
But what I really want you to do is coming out of this stuff,
it's not just yaw, pitch roll.
How many sets of Euler angles are there?
>> 12.
>> 12 actually.
And we're going to look at that.
And what I want you to do is, I want you to be experts in all 12 sets.
Now, you don't have to memorize 12 sets of equations and transformations and
all of that stuff.
But there's clear categorizations of those 12 sets of all the angles,
this grouping behaves this way this grouping behaves this way.
These are the singularities there's three parameters that must be singularities.
What are the differential kinematic equations?
How do we add them?
How do we subtract them?
They're not vectors.
Even though we write yaw pitch role often as a three by one set,
it's not a vector set.
We can't just advert.
So we're going to look at all those different things.
We'll start today.
We're not going to finish today.
We'll going to be wrapping up, specially the additions and that sort of stuff,
the next lecture.
So really, they're very common set and while you hear me
making maybe at times comments that might sound disparaging like the total crap.
People might take that negatively.
They actually do have big values especially in robotic systems.
You often have shoulder links that, this link can move this way.
And then, your arm can twist this way and the last one can do other motion.
And the often mechanical building systems that are sequential rotation sequences
where this math is absolutely perfect.
Where I'm less thrilled with them is three dimensional rotations.
because you'll see they're the most singular set that we have.
You're never further than 90 degrees away from a singularity.
Just kind of bad.
There's much better sets to use, general spacecraft rotations.
But they're a very fundamental set.
For small angles, people are very familiar with yaw, pitch, roll,
kind of aircraft-like things.
So you definitely, this is one of your tools in your Swiss Army knife, and
you really need to know what they are.
Even though it may not be a primary tool for spacecraft dynamics,
you have to understand them and how they come.
So very common set. The other key is sequential so
when Evan was doing his motion.
Really this is where you want to use your right hand again not your left hand.
because everything is right-handed here.
You start out with the original orientation.
And now we're going to specify how we are going to rotate and
the order actually matters.
If we had done the roll, and then the pitch, and then the yaw,
you get a different attitude, and I'll illustrate this in a second.
So that's why the sequence, I always love this when I get these angles and,
here's your and they give me something like this is your roll, pitch, yaw.
Which immediately raises flag for me.
Why are you saying roll, pitch, yaw and not yaw, pitch, roll?
So I asked him and emailed him and said, this is a three, two, one sequence.
I get back the question mark.
What do you mean?
I'm like, aw, crap.
We're in trouble.
Just give me the math.
If I look at the math I know precisely what you're doing.
And some people do do a one two three sequence.
Others have a three, two, one sequence.
They're different.
They may linearize the equivalent but for
large [INAUDIBLE] anything beyond the linear regime, they're not the same thing.
So we really want to be careful.
The way we label them, you see here, I have an IJK.
So this could be three, two, one.
The orbit was a three, one, three.
And there's different combinations.
In fact, we said 12.
For the first axis, let's go through that.
As Maurice was saying we have to, with the all angle sequence we're not rotating
about some arm but through body axis.
We're using a primary one, two and three.
That's why when you define your body-fixed frame this is critical.
What is one?
What is two?
What is three?
Again, space station has lots of body-fixed frames.
So you have to be careful.
Now, how many options do you have for your first choice, first rotation?
>> Three. >> Three.
I can do one, two, or three.
But let's say I picked two just to mix things up.
If I picked two for my second rotation, how many options do I have?
Why two?
>> You would a two axis rotation,
another two axis rotation would be the same as just adding two-
>> Exactly, because then you introduce,
you don't get three degrees of freedom,
you've reused the same degree of freedom twice, sequentially.
That's not giving you any new stuff.
So if you did a two, you can't do two again, but I can do one or three.
So let's pick one, now I get a two, one.
Now, that I did the one rotation, how many options do I have for the last one?
Two again.
I could reuse two again, or I can go three.
If I want to spread out across all the axis.
And that would be some of the fundamental differences.
But it's 3 times 2 times 2, which is 12.
That's how we come up with 12 possible combinations of all the angles.
Okay, good.
Easy to visualize for small rotations.
Again, I can just give you this angles and you can quickly go.
Do the quick twists with your wrists and that must be the attitude.
That's kind of the nice thing about them, somewhat intuitive.
And it's good to know.
If you have large rotations, and you'll see some animations, then it's trickier,
if I tell you hey your yaws 180, your inclination is 1,
and your pitch is minus 140 what's the attitude?
And I really have to go through the complete sequence.
I can't visualize quickly, is that large a departure or not?
because with all our angles, you can have stuff where you move far,
you twitch a little bit, and you almost move all the way back.
Now, you have two very large angles but
actually the angular difference between two frames could be very very small.
That's some of the stuff that you'll see.
So here's my quick spacecraft thing that I drew up, well, 30 years ago.
But this is like an aircraft like frame.
We talked about the third axis points down.
First axis points towards the nose and
the second axis points towards the right wing tip.
This way you put your thumb out there.
That's a positive pitch up, positive yaw to the right and
a positive roll has you dipping to the right.
But this not universal, ships actually have the yaw typically up.
So their pitch warps, and it's different.
Different kind of frames.
So [COUGH] many ways you can do this.
But just use your right hand along the axis.
The sequence.
This is just a mathematical illustration.
So you really go from, originally if you have 40 degrees, 60 degrees,
minus 50, I would have to go, I'm going to rotate 40 degrees,
then I'm going to do a 3 2, that's this 1.
What did I say?
50, 40, that would be down.
And then, the last one, negative this way, and
that would be roughly the final attitude.
But it's a sequence of rotations.
And as we go through this I used these notations a little bit in my development.
As we're doing a sequence, I'm starting out with b and n being identical.
Then, I do my first rotation.
This really creates a new frame.
That's what I call the b prime frame.
Then I do the second rotation, and then that gets me the double prime frame.
And then, I do the final rotation, which could give you the triple prime.
But really, the final rotation is the body orientation.
That is the spacecraft relative to this inertial frame.
But you went through some sequence to get there.
So if you see primes and double primes,
those are those intermediate first two frames, and it'll be useful when
we develop the differential kinematic equations to relate these rotation axises.
Okay, good.
This is, said that was a 3-2-1 sequence we looked at, which is what we have for
aircraft also sometimes a spacecraft.
Orbits has this one to 3-1-3 and really you do your first
a 3 rotation that's about your north pole axis here and
that goes up and that gives you your line of nodes.
Then, we do a one takes a one axis from here to here the one axis now kind of like
one prime, becomes your Axis about which you're doing inclination.
And then now you're in the orbit plane.
Your third axis now normal to the orbit plane, and we rotate about the b3 to
switch up our line of periapsis up to here, it's a 3-1-3 sequence.
So for a 3-1-3 sequence, where is the singularity going to occur?
Exactly. The inclination is zero.
And you notice the inclination happens to be the second angle.
It turns out that across al the Euler angle sets,
all 12 of them, it's always the second angle that's a trouble maker.
This is a symmetric set.
Symmetric means the The first and the third number are the same,
they are repeated.
Alright so we do a three-one-three, two-one-two,
one-two-one, three-one-three, and so forth, that's what we have.
Then, always this angle is zero.
That's one definition for geometric interpretation for singularity.
What's the other singularity we have?
180, right?
In fact, if you flip it.
So that's why you typically see in orbits,
your angles are always defined between 0 and 180.
Otherwise the ascending node is defined between 0 and 360, 0 and 360.
So they move the nodes around
such that your inclinations are always a positive value, never a negative value.
So that kind of clips the range of these coordinates.
You never have minus 10 inclination.
Okay, good dude, this one's easier to visualize.
Anybody remember three to one?
Where there are singularities happen?
>> It should 90 degrees?
>> Yes, second angle and then it's 90, plus or minus.
Again, there's two possible ones and you will see the mathematics here shortly.
But it's true actually for all of the asymmetric sets.
One, two, three.
Three, two, one.
Two, one, three.
Any of those.
It's always going to be the second angle, and plus minus 90.
Whatever's going on in the math.
That's where it goes singular.
So those are some nice patterns.
Here's some quick videos we can see.
I'm showing you essentially a three, two, one sequence of 60,
50, 70, and a three, one, three of 60, 50, 70, just showing that the order matters.
If you rotate by different axis, you get different amounts of final attitudes.
So not too surprising.
But If I keep the same rotation on the left hand side and
the right hand side, I adjust me three-one-three angles to be this.
You can see at the end of the sequential ration sequence, wa-lah,
we lock in, and we have exactly the same orientation.
So in essence this means, how do I convert 60,50,70,
3-2-1 Euler Angels into the equivalent 3-1-3?
This is an essence according to transformation that you have to do.
As like saying, look,
I've gotta cartesian position and then now I have to have equivalent spherical.
We have to be able to translate between all of this different And I'll show you.
In fact, the easiest way to do this is through the direction cosine matrix, and
we'll show that in a second, how we relate those to each other.
But this is just a visualization that you do get there, but
it's different rotation sequences that we do.
Here's another one, where I have a one, three,
two sequence And you can see it has two larger angles and one really small angle.
And that gets you there so again to me this highlights how misleading sometimes
it can be if you have these big angles.
Does that mean you're really far away from where you want to be?
And its not always the case with Euler angles you can move over here,
do a little twitch almost come back And in here in this case,
there's two larger angles, but one of them is almost zero.
So that kind of immediately tells me that it's actually not that far
away from it again.
So the visualization's easy for small angles.
Large ones, I definitely have to go through the sequence And
actually other parameters are easier for me to interpret for large.
Am I close to upside down.
Is this tracking error large or small.
That's something we'll compare to with other coordinate descriptions.
Okay. So
with this definition, two types of Euler angles.
Symmetric set, asymmetric set.
This is something you know I'm going to review next time, so
you might as well review it again.
We're going to go over this.
Each has their own mathematical properties and singularities.
And you will see for one of them, we can do easy additions.
Another one, we pretty much have to go through DCM.
I haven't seen any elegant closed form stuff.
But those are the two types, asymmetric and symmetric.
Singularities, we talked about this one.
For a symmetric set, and I'm using the 3-1-3 as a representation of that.
It's just easier to visualize and people familiar with the orbits.
It's if the Planation\g is 0, 180.
Then you repeat rotations about the n3 or minus n3.
Either way, n3, minus n3, it's still the same axis about which you're rotating.
And that's where you lose the uniqueness issues.
If it's an asymmetric set, it's plus, minus 90 degrees,
which is basically pitch 90 up and down.
A little bit harder to visualize.
But there, too,
you end up having mathematical ambiguities in your mathematics.
Of describing it and extracting these angles.
You can see with zero to 180, the best you can be with 90 degrees, a polar orbit.
And now you're 90 degrees away from a singularity.
That's as far as it goes.
With these, the pitch, you'd be at zero pitch.
That puts you right in the sweet spot between the singular behavior.
But you're never more than 90 degrees away.
Just keep that in mind as we look at other attitude sets, and start to compare.
You got a question?
>> No.
>> No. You're just doing your finger things.
Perfect.
Following along.
Good.
These are the singularities we discussed.
