Possible Expressions for Matrix Rotation Reduction

[Deadstar]

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.

 

[Stephen Tashi]

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

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

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?