I'm wondering if the following is possible.(adsbygoogle = window.adsbygoogle || []).push({});

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

[itex] \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}[/itex]

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 [itex]\theta[/itex] 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

[itex] \begin{pmatrix} a \\ b \\ c \end{pmatrix} = R_1(C_1) R_2(C_1) \begin{pmatrix} p \\ q \\ r \end{pmatrix}[/itex]

Where [itex]C_1[/itex] and [itex]C_2[/itex] are constants, again which rotation matrices are used doesn't really matter.

We could alternatively write this as.

[itex]\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}[/itex]

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

[itex]\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}[/itex]

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

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# Matrix rotation reduction

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