欧拉角和四元数的相互转换_四元数转欧拉角

设四元数为:
$$q=(w,x,y,z)$$

对应欧拉角为:

• 俯仰角(pitch):$\theta$
• 滚转角(roll):$\varphi$
• 偏航角(yaw):$\psi$
  

则四元数到欧拉角的转换公式为:

欧拉角到四元数

w = cos ⁡ ( θ 2 ) cos ⁡ ( ϕ 2 ) cos ⁡ ( ψ 2 ) + sin ⁡ ( θ 2 ) sin ⁡ ( ϕ 2 ) sin ⁡ ( ψ 2 ) x = sin ⁡ ( θ 2 ) cos ⁡ ( ϕ 2 ) cos ⁡ ( ψ 2 ) − cos ⁡ ( θ 2 ) sin ⁡ ( ϕ 2 ) sin ⁡ ( ψ 2 ) y = cos ⁡ ( θ 2 ) sin ⁡ ( ϕ 2 ) cos ⁡ ( ψ 2 ) + sin ⁡ ( θ 2 ) cos ⁡ ( ϕ 2 ) sin ⁡ ( ψ 2 ) z = cos ⁡ ( θ 2 ) cos ⁡ ( ϕ 2 ) sin ⁡ ( ψ 2 ) − sin ⁡ ( θ 2 ) sin ⁡ ( ϕ 2 ) cos ⁡ ( ψ 2 )

$$\begin{aligned} w &= \cos(\frac{\theta}{2})\cos(\frac{\phi}{2})\cos(\frac{\psi}{2}) + \sin(\frac{\theta}{2})\sin(\frac{\phi}{2})\sin(\frac{\psi}{2}) \\ x &= \sin(\frac{\theta}{2})\cos(\frac{\phi}{2})\cos(\frac{\psi}{2}) - \cos(\frac{\theta}{2})\sin(\frac{\phi}{2})\sin(\frac{\psi}{2}) \\ y &= \cos(\frac{\theta}{2})\sin(\frac{\phi}{2})\cos(\frac{\psi}{2}) + \sin(\frac{\theta}{2})\cos(\frac{\phi}{2})\sin(\frac{\psi}{2}) \\ z &= \cos(\frac{\theta}{2})\cos(\frac{\phi}{2})\sin(\frac{\psi}{2}) - \sin(\frac{\theta}{2})\sin(\frac{\phi}{2})\cos(\frac{\psi}{2}) \end{aligned}$$ wxyz​=cos(2θ​)cos(2ϕ​)cos(2ψ​)+sin(2θ​)sin(2ϕ​)sin(2ψ​)=sin(2θ​)cos(2ϕ​)cos(2ψ​)−cos(2θ​)sin(2ϕ​)sin(2ψ​)=cos(2θ​)sin(2ϕ​)cos(2ψ​)+sin(2θ​)cos(2ϕ​)sin(2ψ​)=cos(2θ​)cos(2ϕ​)sin(2ψ​)−sin(2θ​)sin(2ϕ​)cos(2ψ​)​

四元数到欧拉角

θ = arcsin ⁡ ( 2 ( w ⋅ y − z ⋅ x ) ) ϕ = arctan ⁡ 2 ( 2 ( w ⋅ x + y ⋅ z ) , 1 − 2 ( x 2 + y 2 ) ) ψ = arctan ⁡ 2 ( 2 ( w ⋅ z + x ⋅ y ) , 1 − 2 ( y 2 + z 2 ) )

$$\begin{aligned} \theta &= \arcsin(2(w \cdot y - z \cdot x)) \\ \phi &= \arctan2(2(w \cdot x + y \cdot z), 1 - 2(x^2 + y^2)) \\ \psi &= \arctan2(2(w \cdot z + x \cdot y), 1 - 2(y^2 + z^2)) \end{aligned}$$ θϕψ​=arcsin(2(w⋅y−z⋅x))=arctan2(2(w⋅x+y⋅z),1−2(x2+y2))=arctan2(2(w⋅z+x⋅y),1−2(y2+z2))​