KVKV-X

Rotation conversion

Use cases

Robot arms
AGV
Vision sensors
Palletizers

List

Euler angles → rotation matrix

Rotation matrix → Euler angles

Euler angles → rotation matrix (EulerToMatrix)

KV-X

Input

eulerAngle: Euler angles [roll, pitch, yaw]
order: Rotation order (0:XYZ, 1:XZY, 2:YXZ, 3:YZX, 4:ZXY, 5:ZYX)

Output

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}

[Formula]

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)

KV-X

Input

mat: Rotation matrix
order: Rotation order (0:XYZ, 1:XZY, 2:YXZ, 3:YZX, 4:ZXY, 5:ZYX)

Output

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]

[Formula]

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)

KV-X

Input

eulerAngle: Euler angles [roll, pitch, yaw]
order: Rotation order (0:XYZ, 1:XZY, 2:YXZ, 3:YZX, 4:ZXY, 5:ZYX)

Output

qt: Quaternion [w, x, y, z]

[Example]
eulerAngle = [30, 30, 0]
order = 0

qt = [0.933, 0.25, 0.25, 0.067]

[Formula]

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)

KV-X

Input

qt: Quaternion [w, x, y, z]
order: Rotation order (0:XYZ, 1:XZY, 2:YXZ, 3:YZX, 4:ZXY, 5:ZYX)

Output

eulerAngle: Euler angles [roll, pitch, yaw]

[Example]
qt = [0.933, 0.25, 0.25, 0.067]
order = 0

eulerAngle = [30, 30, 0]

[Formula]

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)

KV-X

Input

qt: Quaternion [w, x, y, z]

Output

mat: Rotation matrix

[Example]
qt = [0.933, 0.25, 0.25, 0.067]
order = 0

mat = \begin{bmatrix} 0.866& 0& 0.5 \\ 0.25& 0.866& -0.433 \\ -0.433& 0.5& 0.75 \end{bmatrix}

[Formula]

\[\large \begin{bmatrix} 2q_w^2+2q_x^2-1& 2q_xq_y-2q_zq_w& 2q_xq_z+2q_yq_w \\ 2q_xq_y+2q_zq_w& 2q_w^2+2q_y^2-1& 2q_yq_z-2q_xq_w \\ 2q_xq_z-2q_yq_w& 2q_yq_z+2q_xq_w& 2q_w^2+2q_z^2-1 \\ \end{bmatrix} \]

Rotation matrix → quaternion (MatrixToQuaternion)

KV-X

Input

mat: Rotation matrix
conv: Conversion candidate (0-3)

Output

qt: 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

qt = [0.933, 0.25, 0.25, 0.067]

[Formula]

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 \[\Large q= \begin{bmatrix} \frac{\sqrt{m_{00}+m_{11}+m_{22}+1}}{2} \\ \frac{m_{21}-m_{12}}{2\sqrt{m_{00}+m_{11}+m_{22}+1}} \\ \frac{m_{02}-m_{20}}{2\sqrt{m_{00}+m_{11}+m_{22}+1}} \\ \frac{m_{10}-m_{01}}{2\sqrt{m_{00}+m_{11}+m_{22}+1}} \\ \end{bmatrix} \]