旋转的描述：矩阵 欧拉角 角轴 四元数 Lie Group
一、绕全局直角坐标系旋转
^{B}\boldsymbol{r} & =x \hat{i}+y \hat{j}+z \hat{k} \quad ^{G}\boldsymbol{r} =X \hat{I}+Y \hat{J}+Z \hat{K} \\ \\
^{G}\boldsymbol{r} & =Q \, ^{B}\boldsymbol{r} \; {\color{red}\Rightarrow} \; \begin{pmatrix}
X \\ Y \\Z
\end{pmatrix} = \begin{pmatrix}
\hat{I} \\ \hat{J} \\ \hat{K}
\end{pmatrix} \begin{pmatrix}
\hat{i} & \hat{j} & \hat{k}
\end{pmatrix} \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
\end{align}$$
$$\begin{align}
Q_{Z,\alpha} =\begin{pmatrix}
\cos \alpha & -\sin \alpha & 0 \\
\sin \alpha & \cos \alpha & 0 \\
0 & 0 & 1
\end{pmatrix} \\ \\
Q_{Y,\beta } =\begin{pmatrix}
\cos\beta & 0 & \sin \beta \\
0 & 1& 0 \\
-\sin \beta & 0 & \cos \beta \\
\end{pmatrix} \\ \\
Q_{X,\gamma} =\begin{pmatrix}
1 & 0 & 0 \\
0 & \cos \gamma & -\sin \gamma \\
0 & \sin \gamma & \cos \gamma \\
\end{pmatrix}
\end{align}$$
#### 实际推导
![[test.webm]]
### 二、绕全局直角坐标系连续旋转
$$\begin{align}
\large ^{G}Q_{B}= Q_{X,\gamma}\;Q_{Y,\beta} \;Q_{Z,\alpha}
\end{align}$$
$$\begin{align}
\left(\begin{array}{ccc}
\cos \alpha\,\cos \beta & -\cos \beta\,\sin \alpha & \sin \beta\\
\cos \gamma\,\sin \alpha+\cos \alpha\,\sin \beta\,\sin \gamma & \cos \alpha\,\cos \gamma-\sin \alpha\,\sin \beta\,\sin \gamma & -\cos \beta\,\sin \gamma\\
\sin \alpha\,\sin \gamma-\cos \alpha\,\cos \gamma\,\sin \beta & \cos \alpha\,\sin \gamma+\cos \gamma\,\sin \alpha\,\sin \beta & \cos \beta\,\cos \gamma
\end{array}\right)
\end{align}$$
```MATLAB
syms a b c
Q1=[cos(a),-sin(a),0
sin(a),cos(a),0
0,0,1
]
Q2=[cos(b),0,sin(b)
0,1,0
-sin(b),0,cos(b)]
Q3=[1,0,0
0,cos(c),-sin(c)
0,sin(c),cos(c)]
Q3*Q2*Q1
```
$$\begin{align}
\alpha=\dfrac{2\pi}{n} \quad \quad {\large Q_{X,\alpha}^{m}}
=\begin{pmatrix}
1 & 0 & 0\\
0 & \cos \left(m\dfrac{2\,\pi }{n}\right) & -\sin \left(m\dfrac{2\,\pi }{n}\right)\\
0 & \sin \left(m\dfrac{2\,\pi }{n}\right) & \cos \left(m\dfrac{2\,\pi }{n}\right)
\end{pmatrix}
\end{align}$$
```
syms m n
assume(m, 'positive')
assumeAlso(m, 'integer')
assume(n, 'integer')
k=2*pi/n
Q=[1,0,0
0,cos(k),-sin(k)
0,sin(k),cos(k)]
simplify(Q^m)
```
### 三、全局翻滚角、俯仰角、偏航角
