Derive equations for the roll, pitch, and yaw angles?

Question

How to derive equations for the roll, pitch, and yaw angles corresponding to the rotation matrix R having ijth element rij?

Answer 1

Taking forward-backwards as the $x$-axis and roll is rotations about the $x$- axis \begin{eqnarray*} R_x(r)= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos(r) & \sin(r) \\ 0 & -\sin(r) & \cos(r) \\ \end{array} \right) \end{eqnarray*}

Taking left-right as the $y$-axis and pitch is rotations about the $y$-axis \begin{eqnarray*} R_y(p)= \left( \begin{array}{ccc} \cos(p) & 0 & -\sin(p) \\ 0 & 1 & 0 \\ \sin(p) & 0 & \cos(p) \\ \end{array} \right) \end{eqnarray*}

Taking up-down as the $z$-axis and yaw is rotations about the $z$-axis \begin{eqnarray*} R_z(y) =\left( \begin{array}{ccc} \cos(y) & \sin(y) & 0 \\ -\sin(y) & \cos(y) & 0 \\ 0 & 0 & 1 \end{array}\right) \end{eqnarray*}

Do the matrix multiplication ... \begin{eqnarray*} R_z(y) R_y(p) R_x(r) = \\ \left( \begin{array}{ccc} \cos(y) \cos(p) & \cos(y)\sin(p)\sin(r)+\sin(y)\cos(r) & -\cos(y)\sin(p)\cos(r)+\sin(y)\sin(r) \\ -\sin(y\cos(p)) & -\sin(y)\sin(p)\sin(r)+\cos(y)\cos(r) & \sin(y)\sin(p)\cos(r)+\cos(y)\cos(r) \\ \sin(p) & \sin(r)\cos(p) & \cos(p)\cos(r) \end{array} \right) \end{eqnarray*}

Your rotation matrix should have the structure above. The angles can be deduced from the following formulea \begin{eqnarray*} p =\sin^{-1}( R_{1,3}) \\ r= \cos^{-1}(\frac{R_{3,3}}{\cos(p)}) \\ y= \cos^{-1}(\frac{R_{1,1}}{\cos(p)}) \\ \end{eqnarray*} You should check the other component satisfy this structure.