rotating coordinate systems: example 1

conventions and setup

As far as possible, I am going to stay with my notation. r and \rho are the old and new (fixed and rotating) components of the position vector; v and \nu are derivatives wrt time of r and \rho respectively; a and \alpha are derivatives wrt time of v and \nu respectively. (But R is a convenient scalar value, and will no longer denote the position vector whose components are r and \rho\ .)

v = \dot{r}

\nu = \dot{\rho}

a = \dot{v}

\alpha = \dot{\nu}

The rotating frame is the same in all these problems, so get its matrices early (hence not often). The z-axis is our axis of rotation.

The attitude matrix for a CCW rotation of the axes (about the z-axis) is…

A = \left(\begin{array}{lll} \cos (t \omega ) & \sin (t \omega ) & 0 \\ -\sin (t \omega ) & \cos (t \omega ) & 0 \\ 0 & 0 & 1\end{array}\right)

The transition matrix is… Read the rest of this entry »

rotating coordinate systems: background

I owe you derivations of three assertions. We will need a fourth one, too.

  1. matrix multiplication by N is equivalent to some vector cross product
  2. the transition matrix is T =  1 - \sin (\theta )\ N + (1-\cos (\theta ))\ N^2
  3. \dot{T} = - \omega\ N\ T\
  4. N^3 = -N

matrix multiplication by N

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rotating coordinate systems: equations


There are three key equations for rotational mechanics. Let me refer to them as “the equations”. Goldstein writes a general equation for “some arbitrary vector G”:

\left(\frac{\text{dG}}{\text{dt}}\right)_{\text{space}}=\left(\frac{\text{dG}}{\text{dt}}\right)_{\text{rotating}} + \omega \times G

a specific equation for velocity:

v_s=v_r + \omega \times r

(that’s the first equation with G = position vector r) and an equation for acceleration:

a_s=a_r + 2\ \omega \times v_r +\omega \times (\omega \times r)

Let’s look at all that. I will derive these 3, and a general second-derivative equation, but I will have to return and derive some auxiliary facts.
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Happenings – 21 June

I’m working on several things, and it’s possible I’ll have a technical post ready later today. OTOH, I’m going to a dinner party tonight, so this will be a short schoolday.

As you might guess from the recent posts about rotations, I have gotten caught up in rotating coordinate systems. The original cause was a nifty equation in the airplane control books. As is true of too many things, I can even find that equation in an old schoolbook, in this case my ancient copy of Goldstein. Worse, I highlighted it all those years ago. That equation writes the rotation axis \omega in terms of the derivatives of the Euler angles and their rotation axes.
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books added 21 June

The following books have been added to the bibliography.

The Ashley book is a welcome addition to control of flight vehicles (Bryson; Blakelock): it’s got a lot more detail about the underlying dynamics. I have no idea when I bought it, but I eventually remembered that it was somewhere in my library, and was delighted to find its more detailed explanation – and excellent drawing – of the various coordinate systems in use for aircraft and missles. This is material which the control theory books assume you’ve seen in more detail.

The Ideals & Varieties book is an introductory text which I am working thru with a friend. The third author, O’Shea, is the author of a recent book on the Poincare conjecture which is what got me started on the geometry of surfaces.

The 3 mechanics books (Marion, Symon, and the Berkeley) were additional references (cf. Goldstein) for acceleration in rotating coordinate systems. I have listed the Berkeley text twice, for the same reason I list Schaum’s Outlines twice. I’ve always heard it called “the Berkeley mechanics book”, and that’s how I searched to see if it – and the rest of the series – were in print (no) and available used (yes).

I bought the Basilevsky Factor Analysis book because I wanted something more about noise in factor analysis methods (cf. Malinowski). It looks like a good and interesting book (I wasn’t expecting to find the Kalman filter in it), although it is the specific text in which I found the mistaken assertion that we could always choose the eigenvector matrix orthogonal. As I said when I corrected that very same careless error on one of my own SVD pages, I am inclined to be tolerant of other people’s mistakes: I make mistakes, too.
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PCA / FA Malinowski: Example 5. missing data.

Malinowski does use H for something else, namely missing data points. The X matrix must be complete, but a test vector x need not be.

For quick reference, X, H and u are

X = \left(\begin{array}{lll} 2 & 3 & 4 \\ 1 & 0 & -1 \\ 4 & 5 & 6 \\ 3 & 2 & 1 \\ 6 & 7 & 8\end{array}\right)

H =\left(\begin{array}{lllll} 0.203008 & -0.180451 & 0.218045 & -0.165414 & 0.233083   \\ -0.180451 & 0.327068 & -0.0827068 & 0.424812 & 0.0150376   \\ 0.218045 & -0.0827068 & 0.308271 & 0.0075188 & 0.398496   \\ -0.165414 & 0.424812 & 0.0075188 & 0.597744 & 0.180451   \\ 0.233083 & 0.0150376 & 0.398496 & 0.180451 & 0.56391\end{array}\right)

u = \left(\begin{array}{lllll} 0.327517 & 0.309419 & -0.813733 & 0.257097 & -0.262167   \\ -0.0107664 & -0.571797 & -0.464991 & -0.668451 &   0.0994427 \\ 0.538684 & 0.134501 & 0. & 0. & 0.831703 \\ 0.200401 & -0.746715 & 0. & 0.634172 & -0.00904025 \\ 0.749851 & -0.0404178 & 0.348743 & -0.291376 & -0.479133\end{array}\right)

Let’s try a magic vector, with one missing value, marked NA. This vector came to me in a dream. (Not! But it might as well have. There is no way in the real world I would know this vector.)

x = \{1,\ 2,\ 3,\ \text{NA},\ 5\}

Is there a value of NA which would put this vector in the 2D subspace? (Yes, but I know this because I used this vector to construct the data matrix X!)
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PCA / FA Malinowski: Example 5. Simplified Target Testing


For a good reason which I have not yet discussed, Malinowski wants to find x vectors which are close to \hat{x} = H\ x vectors. (His x and \hat{x} are usually written y and \hat{y} for a least-squares fit). He finds a possible x and tests it to see if xhat is close. He recommended computing an intermediate t vector which was the \beta for his least-squares fit to x.

Since he seems to care about t only when \hat{x} is close to x, and since \hat{x} is incredibly easy to compute directly, I prefer to delay the computation of t. Find t after we’ve found a good x.

It will also turn out that he wants a collection of t vectors in order to pick a nicer basis than u or u1. And I’m not going to follow him there, because all of that is what practitioners call “non-orthogonal rotations”. (That strikes me as an oxymoron.) It’s what Harman spends most of his book doing, and that’s where I’ll look if I ever I want to. It’s important, but I’m not going to look at it this time around.

Anyway, we factored the data matrix
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