3d Rotation about axis (x, y, and z)

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rogue909

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**Information that doesn't matter**
I am working on a robot that is designed to pickup audio signals. The plan is to have the robot move to several locations and record bits of data. By knowing the speed of sound I can determine the difference in distance of the sound collection and from there plot it on a global coordinate to determine the source of the sound.

**Information that does matter**
I have a microphone positioned at 3 different positions from my location sensor, these positions are known. The sensor can tell me the current location (x, y, z) of the robot and the angle off of each axis from a global coordinate system.

My question is; How do I convert the position of the microphones from the sensor's coordinate system to the global coordinate system? I know if the angles remained 0 it would be a simple adding and subtracting problem. If there was a single axis angle change it would be a matrix multiplication. I imagine rotating across 3 axis would also result in a matrix operation but I cant seem to find what that matrix operation would be.

Is there a pre-defined matrix that I would need to multiply by, that takes into account each angle to come up with a final answer? Or is there a way to use the individual axis of rotation matrices to determine the position?
 
Rotation about the x-axis, through angle \(\displaystyle \theta\), is given by the matrix
\(\displaystyle \begin{bmatrix}1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & cos(\theta) \end{bmatrix}\).

Rotation about the y-axis, through angle \(\displaystyle \theta\), is given by the matrix
\(\displaystyle \begin{bmatrix}\cos(\theta) & 0 & -\sin(\theta) \\ 0 & 1 & 0 \\ \sin(\theta) & 0 & cos(\theta)\end{bmatrix}\).

Rotation about the z-axis, through angle \(\displaystyle \theta\), is given by the matrix
\(\displaystyle \begin{bmatrix}\cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1\end{bmatrix}\).
 
HallsofIvy said:
Rotation about the x-axis, through angle \(\displaystyle \theta\), is given by the matrix
\(\displaystyle \begin{bmatrix}1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & cos(\theta) \end{bmatrix}\).

Rotation about the y-axis, through angle \(\displaystyle \theta\), is given by the matrix
\(\displaystyle \begin{bmatrix}\cos(\theta) & 0 & -\sin(\theta) \\ 0 & 1 & 0 \\ \sin(\theta) & 0 & cos(\theta)\end{bmatrix}\).

Rotation about the z-axis, through angle \(\displaystyle \theta\), is given by the matrix
\(\displaystyle \begin{bmatrix}\cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1\end{bmatrix}\).
Click to expand...

I understand this much. But I am rotating among all 3 axis at once. What is the matrix given from rotating through x, y, and z?
 
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