科罗拉多大学波德分校

Kinematics: Describing the Motions of Spacecraft

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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

从本节课中

Static Attitude Determination

This module covers how to take an instantaneous set of observations (sun heading, magnetic field direction, star direction, etc.) and compute a corresponding 3D attitude measure. The attitude determination methods covered include the TRIAD method, Devenport's q-method, QUEST as well as OLAE. The benefits and computation challenges are reviewed for each algorithm.

2: TRIAD Method Definition11:39

2.1: TRIAD Method Numerical Example9:28

3: Wahba's Problem Definition11:39

4: Devenport's q-Method16:58

4.1: Example of Devenport's q-Method7:33

5: QUEST9:23

5.1: Example of QUEST3:23

6: Optimal Linear Attitude Estimator5:09

6.1: Example of OLAE2:07

Optional Review: Attitude Determination14:14

Optional Review: Attitude Estimation Algorithms10:23

与讲师见面

Hanspeter Schaub
Alfred T. and Betty E. Look Professor of Engineering
Department of Aerospace Engineering Sciences

Let me go back,

and we're going to run this through Mathematica just to show you the steps.

The setup, the truth, the measure, it's all the same.

The q parameters I'm going to evaluate,

that's all the same because we need the K matrix still.

So I'm just going to evaluate all those cells.

So now we're saying our initial guess is just going to be the sum of the weights.

And you saw the true answer was very close to 2.

But this isn't quite the true optimal answer yet.

If I evaluate, but if I just would use the first guess, I didn't iterate.

You can see, before we had 1.67 something as an error, now I get 1.70.

I'm only 3/100 of a degree difference by using just that sum of the weights as

the optimal answer, right?

So that shows you quickly that okay, I can do better.

But man, even with the first guess, I'm pretty darn close.

That's exciting, so let's iterate for this.

This is my determinant thing, so I made a function out of that so I can call it.

And the derivative of that function with respect to this s value,

that's the eigenvalues, right, that we have of this.

This is where you find your classic stuff.

And so let's just step through this.

That's the function, a negative function.

My initial one is just equal to that sum of the weights, that's equal to 2.

And we saw, that got us to within a few hundredths of a degree to the answer.

So I'm evaluating this, this f function,

if 2 was the root of the determinant, this would have been 0.

I'm close to 0, it's in the thousands, 10 to the minus 3, but

I'm not quite there, right?

Now I'm doing an adjustment with a Newton Rathsman method, and

you can see after one adjustment, man, I'm only showing you some finite digits here.

This starts to look pretty darn close to what we've seen with the eigenvalue

eigenvector problem.

And sure enough,

my differences from 0 jump from 10 to the minus 3 to 10 to the minus 6.

So one correction gave me three orders of magnitude better.

Most applications are already done after one iteration.

This is why it's so much faster than an eigenvalue eigenvector.

It only takes a very small number of iterations.

But we're not happy with 10 to the minus 6, are we?

So we're going to do better.

Here, I'm not showing enough values, but you can see with a second iteration,

I go from 10 to the minus 6 to 10 to the minus 13.

How many problems in life do you have that converge this nicely this quickly?

That is pretty darn good.

And so you can see here, I'll do it one more time, because why not?

Now I'm down to machine precision.

That's as good as it gets.

So this is a really, really fast way to iterate with that good initial guess.

And anyway, and that would give you, if I plug that one back in, I would get exactly

the same answer as Davenport's Q method to machine precision, right?

But it's a much faster way to get there.

Good, so we have two ways to solve Waba's problem,

Davenport's Q method, eigenvalue eigenvector, QUEST method.

It has a good initial guess for that optimal eigenvalue and

then an iterative way to get it, and then you have a closed form answer to get CRPs.

And there are some rotations and things you have to consider to implement it.

But it's still way, way faster.