卡尔丹旋转规则

Cardanian rotation matrix推导（卡尔丹旋转规则）：

按照Yaw(偏转角)->Pitch(翻滚角)->Roll(滚动角)的顺序进行旋转
$Yaw:\kappa ⇢旋转矩阵R_Z(\kappa )$
$Pitch:\varphi ⇢旋转矩阵R_Y(\varphi )$
$Yaw:\omega ⇢旋转矩阵R_X(\omega )$

旋转图示如下：

根据推导，存在：
R Z ( κ ) = [ c o s κ s i n κ 0 − s i n κ c o s κ 0 0 0 1 ] R_Z(\kappa)= [cosκsinκ0−sinκcosκ0001]

RZ​(κ)=⎣ ⎡​cosκ−sinκ0​sinκcosκ0​001​⎦ ⎤​
$$R_Y(\varphi )=\left[\begin{array}{ccc}cos\varphi & 0& -sin\varphi \\ 0& 1& 0\\ sin\varphi & 0& cos\varphi \end{array}\right]$$
$$R_X(\omega )=\left[\begin{array}{ccc}1& 0& 0\\ 0& cos\omega & sin\omega \\ 0& -sin\omega & cos\omega \end{array}\right]$$

按照卡尔丹旋转规则，存在旋转矩阵：
$$R=R_Z(\kappa )R_Y(\varphi )R_X(\omega )$$
R = [ cos ⁡ ( κ ) sin ⁡ ( κ ) 0 − sin ⁡ ( κ ) cos ⁡ ( κ ) 0 0 0 1 ] [ cos ⁡ ( ϕ ) 0 − sin ⁡ ( ϕ ) 0 1 0 sin ⁡ ( ϕ ) 0 cos ⁡ ( ϕ ) ] [ 1 0 0 0 cos ⁡ ( ω ) sin ⁡ ( ω ) 0 − sin ⁡ ( ω ) cos ⁡ ( ω ) ] R = [cos(κ)sin(κ)0−sin(κ)cos(κ)0001]

[cos(ϕ)0−sin(ϕ)010sin(ϕ)0cos(ϕ)] [1000cos(ω)sin(ω)0−sin(ω)cos(ω)] R=⎣ ⎡​cos(κ)−sin(κ)0​sin(κ)cos(κ)0​001​⎦ ⎤​⎣ ⎡​cos(ϕ)0sin(ϕ)​010​−sin(ϕ)0cos(ϕ)​⎦ ⎤​⎣ ⎡​100​0cos(ω)−sin(ω)​0sin(ω)cos(ω)​⎦ ⎤​
$$R=\left[\begin{array}{ccc}\cos (\kappa )\cos (\varphi )& \sin (\kappa )& -\cos (\kappa )\sin (\varphi )\\ -\sin (\kappa )\cos (\varphi )& \cos (\kappa )& \sin (\kappa )\sin (\varphi )\\ \sin (\varphi )& 0& \cos (\varphi )\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos (\omega )& \sin (\omega )\\ 0& -\sin (\omega )& \cos (\omega )\end{array}\right]$$
$$R=\left[\begin{array}{ccc}\cos (\kappa )\cos (\varphi )& \sin (\kappa )\cos (\omega )+\cos (\kappa )\sin (\varphi )\sin (\omega )& \sin (\kappa )\sin (\omega )-\cos (\kappa )\sin (\varphi )\cos (\omega )\\ -\sin (\kappa )\cos (\varphi )& \cos (\kappa )\cos (\omega )-\sin (\kappa )\sin (\varphi )\sin (\omega )& \cos (\kappa )\sin (\omega )+\sin (\kappa )\sin (\varphi )\cos (\omega )\\ \sin (\varphi )& -\cos (\varphi )\sin (\omega )& \cos (\varphi )\cos (\omega )\end{array}\right]$$
$当\kappa ,\varphi ,\omega 很小时，存在:$
$$\begin{gathered} \cos (\kappa )\approx 1.0,\cos (\varphi )\approx 1.0,\cos (\omega )\approx 1.0 \\ \sin (\kappa )\approx \kappa ,\sin (\varphi )\approx \varphi ,\sin (\omega )\approx \omega \\ \varphi \omega \approx 0.0,\kappa \omega \approx 0.0,\kappa \varphi \omega \approx 0.0,\kappa \varphi \approx 0.0 \end{gathered}$$
$则旋转矩阵为:$
$$R=\left[\begin{array}{ccc}1& \kappa & -\varphi \\ -\kappa & 1& \omega \\ \varphi & -\omega & 1\end{array}\right]$$

参考：
[1] Harvey B R . Transformation of 3D Co-ordinates[J]. Australian Surveyor, 1986, 33(2):105-125.