Matrix rotation reduction

1. May 1, 2013

I'm wondering if the following is possible.

Consider some inertial coordiante system x, y, z, and a rotating coordiante system p, q, r defined through matrix rotations as follows.

$\begin{pmatrix} p \\ q \\ r \end{pmatrix} = R_1(\theta_1(t)) R_2(\theta_2(t)) R_3(\theta_3(t)) \begin{pmatrix} x \\ y \\ z \end{pmatrix}$

Where this is simple a 1-2-3 rotation as described in the http://en.wikipedia.org/wiki/Rotation_matrix "In three dimensions" section, with the $\theta$ values being time dependant angles. I used 1,2,3 instead of x,y,z since I already have them as variables and 1,2,3 is more general to arbitrary coordiante systems. The above then is a 1-2-3 rotation but really any combination of rotation matrices can be used, this is just an example.

So let's now define a new coordinate system a,b,c, such that

$\begin{pmatrix} a \\ b \\ c \end{pmatrix} = R_1(C_1) R_2(C_1) \begin{pmatrix} p \\ q \\ r \end{pmatrix}$

Where $C_1$ and $C_2$ are constants, again which rotation matrices are used doesn't really matter.

We could alternatively write this as.

$\begin{pmatrix} a \\ b \\ c \end{pmatrix} = R_1(C_1) R_2(C_1) R_1(\theta_1(t)) R_2(\theta_2(t)) R_3(\theta_3(t)) \begin{pmatrix} x \\ y \\ z \end{pmatrix}$

To show the relationship between a,b,c and our inertial coordinate system.

Now my question is, is there anyway of expressing the above as

$\begin{pmatrix} a \\ b \\ c \end{pmatrix} = R_i(\phi_1(t)) R_j(\phi_2(t)) R_k(\phi_3(t)) \begin{pmatrix} x \\ y \\ z \end{pmatrix}$

Where i, j, k can be 1,2 or 3 to denote which rotation matrix, and $\phi$ time dependant angles that will surely depends on the $\theta$ angles and the constants.

I would have thought yes since a,b,c's orientation is constant with respect to p,q,r, but I'm not sure what the above form would be.

Thanks for any info.

2. May 4, 2013

Stephen Tashi

Can't you do it the trivial way?:

$R_i(\phi_1(t)) = R_1(C_1) R_2(C_1) R_1(\theta_1(t))$
$R_j(\phi_2(t)) = R_2(\theta_2(t))$
$R_k(\phi_3(t)) = R_3(\theta_3(t))$

Or are you defining "rotation matrix" in some way so that the product of rotations matrices is not necessarily a rotation matrix?