Converting Between Euler Angles, Rotation Matrices and Quaternions
This page explains how to convert between Euler angles, rotation matrices and quaternions. It shows formulas for each rotation order (XYZ, XZY, YXZ, YZX, ZXY, ZYX) and usage examples of the PLC library.
Euler angles → rotation matrix (EulerToMatrix)
- eulerAngle
- Euler angles [roll, pitch, yaw]
- order
- Rotation order (0:XYZ, 1:XZY, 2:YXZ, 3:YZX, 4:ZXY, 5:ZYX)
- mat
- Rotation matrix
Example
- eulerAngle
- [30, 30, 0]
- order
- 0
- mat
- \begin{bmatrix} 0.866& 0& 0.5 \\ 0.25& 0.866& -0.433 \\ -0.433& 0.5& 0.75 \end{bmatrix}
Formulas
Rotation order XYZ
\begin{align}
&R_{xyz}=R_xR_yR_z \\
&=
\begin{bmatrix}
cos\theta_y cos\theta_z & -cos\theta_y sin\theta_z & sin\theta_y \\
sin\theta_x sin\theta_y cos\theta_z+cos\theta_x sin\theta_z & -sin\theta_x sin\theta_y sin\theta_z+cos\theta_xcos\theta_z & -sin\theta_x cos\theta_y \\
-cos\theta_x sin\theta_y cos\theta_z+sin\theta_x sin\theta_z & cos\theta_x sin\theta_y sin\theta_z+sin\theta_x cos\theta_z & cos\theta_x cos\theta_y
\end{bmatrix}
\end{align}
Rotation order XZY
\begin{align}
&R_{xzy}=R_xR_zR_y \\
&=
\begin{bmatrix}
cos\theta_y cos\theta_z & -sin\theta_z & sin\theta_y cos\theta_z \\
cos\theta_x cos\theta_y sin\theta_z+sin\theta_x sin\theta_y & cos\theta_x cos\theta_z & cos\theta_x sin\theta_y sin\theta_z-sin\theta_x cos\theta_y \\
sin\theta_x cos\theta_y sin\theta_z-cos\theta_x sin\theta_y & sin\theta_x cos\theta_z & sin\theta_x sin\theta_y sin\theta_z+cos\theta_x cos\theta_y
\end{bmatrix}
\end{align}
Rotation order YXZ
\begin{align}
&R_{yxz}=R_yR_xR_z \\
&=
\begin{bmatrix}
sin\theta_x sin\theta_y sin\theta_z+cos\theta_y cos\theta_z & sin\theta_x sin\theta_y sin\theta_z-cos\theta_y sin\theta_z & cos\theta_x sin\theta_y \\
cos\theta_x sin\theta_z & cos\theta_x cos\theta_z & -sin\theta_z \\
sin\theta_x cos\theta_y sin\theta_z-sin\theta_y cos\theta_z & sin\theta_x cos\theta_y cos\theta_z+sin\theta_y sin\theta_z & cos\theta_x cos\theta_y
\end{bmatrix}
\end{align}
Rotation order YZX
\begin{align}
&R_{yzx}=R_yR_zR_x \\
&=
\begin{bmatrix}
cos\theta_y cos\theta_z & -cos\theta_x cos\theta_y sin\theta_z+sin\theta_x sin\theta_y & sin\theta_x cos\theta_y sin\theta_z+cos\theta_x sin\theta_y \\
sin\theta_x & cos\theta_x cos\theta_z & -sin\theta_x cos\theta_z \\
-sin\theta_y cos\theta_z & cos\theta_x sin\theta_y sin\theta_z+sin\theta_x cos\theta_y & -sin\theta_x sin\theta_y sin\theta_z+cos\theta_x cos\theta_y
\end{bmatrix}
\end{align}
Rotation order ZXY
\begin{align}
&R_{zxy}=R_zR_xR_y \\
&=
\begin{bmatrix}
-sin\theta_x sin\theta_y sin\theta_z+cos\theta_y cos\theta_z & -cos\theta_x sin\theta_z & sin\theta_x cos\theta_y sin\theta_z+sin\theta_y cos\theta_z \\
sin\theta_x sin\theta_y cos\theta_z+cos\theta_y sin\theta_z & cos\theta_x cos\theta_z & -sin\theta_x cos\theta_y cos\theta_z+sin\theta_y sin\theta_z \\
-cos\theta_x sin\theta_y & sin\theta_x & cos\theta_x cos\theta_y
\end{bmatrix}
\end{align}
Rotation order ZYX
\begin{align}
&R_{zyx}=R_zR_yR_x \\
&=
\begin{bmatrix}
cos\theta_y cos\theta_z & sin\theta_x sin\theta_y cos\theta_z-cos\theta_x sin\theta_z & cos\theta_x sin\theta_y cos\theta_z+sin\theta_x sin\theta_z \\
cos\theta_y sin\theta_z & sin\theta_x sin\theta_y sin\theta_z+cos\theta_x cos\theta_z & cos\theta_x sin\theta_y sin\theta_z+sin\theta_x cos\theta_z \\
-sin\theta_y & sin\theta_x cos\theta_y & cos\theta_x cos\theta_y
\end{bmatrix}
\end{align}
Rotation matrix → Euler angles (MatrixToEuler)
- mat
- Rotation matrix
- order
- Rotation order (0:XYZ, 1:XZY, 2:YXZ, 3:YZX, 4:ZXY, 5:ZYX)
- eulerAngle
- Euler angles [roll, pitch, yaw]
Example
- mat
- \begin{bmatrix} 0.866& 0& 0.5 \\ 0.25& 0.866& -0.433 \\ -0.433& 0.5& 0.75 \end{bmatrix}
- order
- 0
- eulerAngle
- [30, 30, 0]
Formulas
The elements of the rotation matrix are written as follows
\[
\begin{bmatrix}
m_{00}& m_{01}& m_{02} \\
m_{10}& m_{11}& m_{12} \\
m_{20}& m_{21}& m_{22} \\
\end{bmatrix}
\]
Rotation order XYZ
\[\large
\begin{align}
\theta_x &= \begin{cases}
\arctan \left( -\frac{m_{12}}{m_{22}} \right) &(\cos \theta_y \neq 0) \\
\arctan \left( \frac{m_{21}}{m_{11}} \right) &(otherwise)
\end{cases} \\
\theta_y &= \arcsin (m_{02}) \\
\theta_z &= \begin{cases}
\arctan \left( -\frac{m_{01}}{m_{00}} \right) &(\cos \theta_y \neq 0) \\
0 &(otherwise)
\end{cases} \\
\end{align}
\]
Rotation order XZY
\[\large
\begin{align}
\theta_x &= \begin{cases}
\arctan \left( \frac{m_{21}}{m_{11}} \right) &(\cos \theta_z \neq 0) \\
\arctan \left( -\frac{m_{12}}{m_{22}} \right) &(otherwise)
\end{cases} \\
\theta_y &= \begin{cases}
\arctan \left( \frac{m_{02}}{m_{00}} \right) &(\cos \theta_z \neq 0) \\
0 &(otherwise)
\end{cases} \\
\theta_z &= \arcsin (-m_{01}) \\
\end{align}
\]
Rotation order YXZ
\[\large
\begin{align}
\theta_x &= \arcsin (-m_{12}) \\
\theta_y &= \begin{cases}
\arctan \left( \frac{m_{02}}{m_{22}} \right) &(\cos \theta_x \neq 0) \\
\arctan \left( -\frac{m_{20}}{m_{00}} \right) &(otherwise)
\end{cases} \\
\theta_z &= \begin{cases}
\arctan \left( \frac{m_{10}}{m_{11}} \right) &(\cos \theta_x \neq 0) \\
0 &(otherwise)
\end{cases} \\
\end{align}
\]
Rotation order YZX
\[\large
\begin{align}
\theta_x &= \begin{cases}
\arctan \left( -\frac{m_{12}}{m_{11}} \right) &(\cos \theta_z \neq 0) \\
0 &(otherwise)
\end{cases} \\
\theta_y &= \begin{cases}
\arctan \left( -\frac{m_{20}}{m_{00}} \right) &(\cos \theta_z \neq 0) \\
\arctan \left( \frac{m_{02}}{m_{22}} \right) &(otherwise) \\
\end{cases} \\
\theta_z &= \arcsin (m_{10}) \\
\end{align}
\]
Rotation order ZXY
\[\large
\begin{align}
\theta_x &= \arcsin (m_{21}) \\
\theta_y &= \begin{cases}
\arctan \left( -\frac{m_{20}}{m_{22}} \right) &(\cos \theta_x \neq 0) \\
0 &(otherwise)
\end{cases} \\
\theta_z &= \begin{cases}
\arctan \left( -\frac{m_{01}}{m_{11}} \right) &(\cos \theta_x \neq 0) \\
\arctan \left( \frac{m_{10}}{m_{00}} \right) &(otherwise) \\
\end{cases} \\
\end{align}
\]
Rotation order ZYX
\[\large
\begin{align}
\theta_x &= \begin{cases}
\arctan \left( \frac{m_{21}}{m_{22}} \right) &(\cos \theta_y \neq 0) \\
0 &(otherwise)
\end{cases} \\
\theta_y &= \arcsin (-m_{20}) \\
\theta_z &= \begin{cases}
\arctan \left( \frac{m_{10}}{m_{00}} \right) &(\cos \theta_y \neq 0) \\
\arctan \left( -\frac{m_{01}}{m_{11}} \right) &(otherwise) \\
\end{cases} \\
\end{align}
\]
Euler angles → quaternion (EulerToQuaternion)
- eulerAngle
- Euler angles [roll, pitch, yaw]
- order
- Rotation order (0:XYZ, 1:XZY, 2:YXZ, 3:YZX, 4:ZXY, 5:ZYX)
- q
- Quaternion [w, x, y, z]
Example
- eulerAngle
- [30, 30, 0]
- order
- 0
- q
- q.w=0.933
q.x=0.25
q.y=0.25
q.z=0.067
Formulas
A quaternion is written as follows
\[
q =
\begin{bmatrix}
q_w \\
q_x \\
q_y \\
q_z \\
\end{bmatrix}
\]
Rotation order XYZ
\[\large
\begin{bmatrix}
-\sin \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
\cos \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \sin \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
-\sin \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
\cos \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \sin \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
\end{bmatrix}
\]
Rotation order XZY
\[\large
\begin{bmatrix}
\sin \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
-\cos \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \sin \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
\cos \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\cos \frac{\theta_z}{2} - \sin \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\sin \frac{\theta_z}{2} \\
\sin \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\cos \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\sin \frac{\theta_z}{2} \\
\end{bmatrix}
\]
Rotation order YXZ
\[\large
\begin{bmatrix}
\sin \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
\cos \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \sin \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
-\sin \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
\cos \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\sin \frac{\theta_z}{2} - \sin \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
\end{bmatrix}
\]
Rotation order YZX
\[\large
\begin{bmatrix}
-\sin \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
\sin \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\cos \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\sin \frac{\theta_z}{2} \\
\sin \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
-\sin \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\cos \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\sin \frac{\theta_z}{2} \\
\end{bmatrix}
\]
Rotation order ZXY
\[\large
\begin{bmatrix}
-\sin \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
-\cos \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \sin \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
\cos \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\cos \frac{\theta_z}{2} + \sin \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\sin \frac{\theta_z}{2} \\
\sin \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\cos \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\sin \frac{\theta_z}{2} \\
\end{bmatrix}
\]
Rotation order ZYX
\[\large
\begin{bmatrix}
\sin \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
\sin \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\cos \frac{\theta_z}{2} - \cos \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\sin \frac{\theta_z}{2} \\
\sin \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\sin \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\cos \frac{\theta_z}{2} \\
-\sin \frac{\theta_x}{2}\sin \frac{\theta_y}{2}\cos \frac{\theta_z}{2} + \cos \frac{\theta_x}{2}\cos \frac{\theta_y}{2}\sin \frac{\theta_z}{2} \\
\end{bmatrix}
\]
Quaternion → Euler angles (QuaternionToEuler)
- q
- Quaternion [w, x, y, z]
- order
- Rotation order (0:XYZ, 1:XZY, 2:YXZ, 3:YZX, 4:ZXY, 5:ZYX)
- eulerAngle
- Euler angles [roll, pitch, yaw]
Example
- q
- q.w=0.933
q.x=0.25
q.y=0.25
q.z=0.067 - order
- 0
- eulerAngle
- [30, 30, 0]
Formulas
A quaternion is written as follows
\[
q =
\begin{bmatrix}
q_w \\
q_x \\
q_y \\
q_z \\
\end{bmatrix}
\]
Rotation order XYZ
\[\large
\begin{align}
\theta_x &= \begin{cases}
\arctan \left( -\frac{2q_yq_z-2q_xq_w}{2q_w^2+2q_z^2-1} \right) &(\cos \theta_y \neq 0) \\
\arctan \left( \frac{2q_yq_z+2q_xq_w}{2q_w^2+2q_y^2-1} \right) &(otherwise)
\end{cases} \\
\theta_y &= \arcsin (2q_xq_z+2q_yq_w) \\
\theta_z &= \begin{cases}
\arctan \left( -\frac{2q_xq_y-2q_zq_w}{2q_w^2+2q_x^2-1} \right) &(\cos \theta_y \neq 0) \\
0 &(otherwise)
\end{cases} \\
\end{align}
\]
Rotation order XZY
\[\large
\begin{align}
\theta_x &= \begin{cases}
\arctan \left( \frac{2q_yq_z+2q_xq_w}{2q_w^2+2q_y^2-1} \right) &(\cos \theta_z \neq 0) \\
\arctan \left( -\frac{2q_yq_z-2q_xq_w}{2q_w^2+2q_z^2-1} \right) &(otherwise) \\
\end{cases} \\
\theta_y &= \begin{cases}
\arctan \left( \frac{2q_xq_z+2q_yq_w}{2q_w^2+2q_x^2-1} \right) &(\cos \theta_z \neq 0) \\
0 &(otherwise)
\end{cases} \\
\theta_z &= \arcsin (-(2q_xq_y-2q_zq_w)) \\
\end{align}
\]
Rotation order YXZ
\[\large
\begin{align}
\theta_x &= \arcsin (-(2q_yq_z-2q_xq_w)) \\
\theta_y &= \begin{cases}
\arctan \left( \frac{2q_xq_z+2q_yq_w}{2q_w^2+2q_z^2-1} \right) &(\cos \theta_x \neq 0) \\
\arctan \left( -\frac{2q_xq_z-2q_yq_w}{2q_w^2+2q_z^2-1} \right) &(otherwise) \\
\end{cases} \\
\theta_z &= \begin{cases}
\arctan \left( \frac{2q_xq_y+2q_zq_w}{2q_w^2+2q_y^2-1} \right) &(\cos \theta_x \neq 0) \\
0 &(otherwise)
\end{cases} \\
\end{align}
\]
Rotation order YZX
\[\large
\begin{align}
\theta_x &= \begin{cases}
\arctan \left( -\frac{2q_yq_z-2q_xq_w}{2q_w^2+2q_y^2-1} \right) &(\cos \theta_z \neq 0) \\
0 &(otherwise)
\end{cases} \\
\theta_y &= \begin{cases}
\arctan \left( -\frac{2q_xq_z-2q_yq_w}{2q_w^2+2q_x^2-1} \right) &(\cos \theta_z \neq 0) \\
\arctan \left( \frac{2q_xq_z+2q_yq_w}{2q_w^2+2q_z^2-1} \right) &(otherwise) \\
\end{cases} \\
\theta_z &= \arcsin (-(2q_xq_y+2q_zq_w)) \\
\end{align}
\]
Rotation order ZXY
\[\large
\begin{align}
\theta_x &= \arcsin (2q_yq_z+2q_xq_w) \\
\theta_y &= \begin{cases}
\arctan \left( -\frac{2q_xq_z-2q_yq_w}{2q_w^2+2q_z^2-1} \right) &(\cos \theta_x \neq 0) \\
0 &(otherwise)
\end{cases} \\
\theta_z &= \begin{cases}
\arctan \left( -\frac{2q_xq_y-2q_zq_w}{2q_w^2+2q_y^2-1} \right) &(\cos \theta_x \neq 0) \\
\arctan \left( \frac{2q_xq_y+2q_zq_w}{2q_w^2+2q_x^2-1} \right) &(otherwise) \\
\end{cases} \\
\end{align}
\]
Rotation order ZYX
\[\large
\begin{align}
\theta_x &= \begin{cases}
\arctan \left( \frac{2q_yq_z+2q_xq_w}{2q_w^2+2q_z^2-1} \right) &(\cos \theta_y \neq 0) \\
0 &(otherwise)
\end{cases} \\
\theta_y &= \arcsin (-(2q_xq_z-2q_yq_w)) \\
\theta_z &= \begin{cases}
\arctan \left( \frac{2q_xq_y+2q_zq_w}{2q_w^2+2q_x^2-1} \right) &(\cos \theta_y \neq 0) \\
\arctan \left( -\frac{2q_xq_y-2q_zq_w}{2q_w^2+2q_y^2-1} \right) &(otherwise) \\
\end{cases} \\
\end{align}
\]
Quaternion → rotation matrix (QuaternionToMatrix)
- q
- Quaternion [w, x, y, z]
- mat
- Rotation matrix
Example
- q
- q.w=0.933
q.x=0.25
q.y=0.25
q.z=0.067
- mat
- \begin{bmatrix} 0.866& 0& 0.5 \\ 0.25& 0.866& -0.433 \\ -0.433& 0.5& 0.75 \end{bmatrix}
Formulas
\[\large \begin{bmatrix} 2q_w^2+2q_x^2-1& 2q_xq_y-2q_zq_w& 2q_xq_z+2q_yq_w \\ 2q_xq_y+2q_zq_w& 2q_w^2+2q_y^2-1& 2q_yq_z-2q_xq_w \\ 2q_xq_z-2q_yq_w& 2q_yq_z+2q_xq_w& 2q_w^2+2q_z^2-1 \\ \end{bmatrix} \]
Rotation matrix → quaternion (MatrixToQuaternion)
- mat
- Rotation matrix
- conv
- Conversion candidate (0-3)
- q
- Quaternion [w, x, y, z]
Example
- mat
- \begin{bmatrix} 0.866& 0& 0.5 \\ 0.25& 0.866& -0.433 \\ -0.433& 0.5& 0.75 \end{bmatrix}
- conv
- 0
- q
- q.w=0.933
q.x=0.25
q.y=0.25
q.z=0.067
Formulas
conv=0 \[\Large q= \begin{bmatrix} \frac{m_{21}-m_{12}}{2\sqrt{m_{00}-m_{11}-m_{22}+1}} \\ \frac{\sqrt{m_{00}-m_{11}-m_{22}+1}}{2} \\ \frac{m_{10}-m_{01}}{2\sqrt{m_{00}-m_{11}-m_{22}+1}} \\ \frac{m_{02}-m_{20}}{2\sqrt{m_{00}-m_{11}-m_{22}+1}} \\ \end{bmatrix} \] conv=1 \[\Large q= \begin{bmatrix} \frac{m_{02}-m_{20}}{2\sqrt{-m_{00}+m_{11}-m_{22}+1}} \\ \frac{m_{10}-m_{01}}{2\sqrt{-m_{00}+m_{11}-m_{22}+1}} \\ \frac{\sqrt{-m_{00}+m_{11}-m_{22}+1}}{2} \\ \frac{m_{21}-m_{12}}{2\sqrt{-m_{00}+m_{11}-m_{22}+1}} \\ \end{bmatrix} \] conv=2 \[\Large q= \begin{bmatrix} \frac{m_{10}-m_{01}}{2\sqrt{-m_{00}-m_{11}+m_{22}+1}} \\ \frac{m_{02}-m_{20}}{2\sqrt{-m_{00}-m_{11}+m_{22}+1}} \\ \frac{m_{21}-m_{12}}{2\sqrt{-m_{00}-m_{11}+m_{22}+1}} \\ \frac{\sqrt{-m_{00}-m_{11}+m_{22}+1}}{2} \\ \end{bmatrix} \] conv=3 \[\Large q= \begin{bmatrix} \frac{\sqrt{m_{00}+m_{11}+m_{22}+1}}{2} \\ \frac{m_{21}-m_{12}}{2\sqrt{m_{00}+m_{11}+m_{22}+1}} \\ \frac{m_{02}-m_{20}}{2\sqrt{m_{00}+m_{11}+m_{22}+1}} \\ \frac{m_{10}-m_{01}}{2\sqrt{m_{00}+m_{11}+m_{22}+1}} \\ \end{bmatrix} \]
FAQ
What is the difference between Euler angles and quaternions?
Euler angles are intuitive but suffer from gimbal lock. Quaternions are widely used in attitude control and robot control.
Why does rotation order matter?
Because XYZ and ZYX produce different resulting orientations.
Reference URL
Qiita: conversion between rotation matrices, quaternions and Euler angles (Japanese) https://qiita.com/aa_debdeb/items/3d02e28fb9ebfa357eaf