多旋翼建模之欧拉角微分

本文来自于zing的博文硬核推导-欧拉角微分方程

首先，为了后续推导方便；我们将pitch简写为p ,roll简写为r,yaw简写为y，旋转矩阵可写为

$${\mathbf{C}}_b^n=\left[\begin{array}{ccc}\cos p\cos y& -\cos r\sin y+\sin r\sin p\cos y& \sin r\sin y+\cos r\sin p\cos y\\ \cos p\sin y& \cos r\cos y+\sin r\sin p\sin y& -\sin r\cos y+\cos r\sin p\sin y\\ -\sin p& \sin r\cos p& \cos r\cos p\end{array}\right]$$

$$\frac{d\left({\mathbf{C}}_b^n\right)}{dt}={\mathbf{C}}_b^n=\left[\begin{array}{ccc}-\dot{p}c(y)s(p)-\dot{y}c(p)s(y)& \dot{r}s(r)s(y)-\dot{y}c(r)c(y)+\dot{p}c(p)c(y)s(r)+\dot{r}c(r)c(y)s(p)-\dot{y}s(p)s(r)s(y)& \dot{r}c(r)s(y)+\dot{y}c(y)s(r)+\dot{p}c(p)c(r)c(y)-\dot{r}c(y)s(p)s(r)-\dot{y}c(r)s(p)s(y)\\ \dot{y}c(p)c(y)-\dot{p}s(p)s(y)& \dot{p}c(p)s(r)s(y)-\dot{y}c(r)s(y)-\dot{r}c(y)s(r)+\dot{r}c(r)s(p)s(y)+\dot{y}c(y)s(p)s(r)& \dot{y}s(r)s(y)-\dot{r}c(r)c(y)+\dot{p}c(p)c(r)s(y)+\dot{y}c(r)c(y)s(p)-\dot{r}s(p)s(r)s(y)\\ -\dot{p}c(p)& \dot{r}c(p)c(r)-\dot{p}s(p)s(r)& -\dot{p}c(r)s(p)-\dot{r}c(p)s(r)\end{array}\right]$$

上式可写为：

$${\dot{\mathbf{C}}}_b^n={\mathbf{C}}_b^n{\left[{}^b\mathbf{\omega }\right]}_{\times }$$

其中$${\left[{}^b\omega \right]}_{\times }=\left[\begin{array}{ccc}0& \dot{p}s(r)-\dot{y}c(p)c(r)& \dot{p}c(r)+\dot{y}c(p)s(r)\\ \dot{y}c(p)c(r)-\dot{p}s(r)& 0& \dot{y}s(p)-dr\\ -\dot{p}c(r)-\dot{y}c(p)s(r)& dr-\dot{y}s(p)& 0\end{array}\right]$$

即可写成

$${}^b\omega =\left[\begin{array}{c}\dot{r}-\dot{y}s(p)\\ \dot{p}c(r)+\dot{y}c(p)s(r)\\ -\dot{p}s(r)+\dot{y}c(p)c(r)\end{array}\right]$$

所以

$$\left[\begin{array}{c}{\omega }_x\\ {\omega }_y\\ {\omega }_z\end{array}\right]=\left[\begin{array}{c}\dot{r}-\dot{y}s(p)\\ \dot{p}c(r)+\dot{y}c(p)s(r)\\ -\dot{p}s(r)+\dot{y}c(p)c(r)\end{array}\right]$$

上式可写为：

$$\left[\begin{array}{c}{\omega }_x\\ {\omega }_y\\ {\omega }_z\end{array}\right]=\left[\begin{array}{ccc}1& 0& -s(r)\\ 0& c(r)& s(r)c(p)\\ 0& -s(r)& c(r)c(p)\end{array}\right]\left[\begin{array}{c}\dot{r}\\ \dot{p}\\ \dot{y}\end{array}\right]$$

进一步得到欧拉角的微分形式：

$$\left[\begin{array}{c}\dot{r}\\ \dot{p}\\ \dot{y}\end{array}\right]=\left[\begin{array}{ccc}1& \tan \theta \sin \varphi & \tan \theta \cos \varphi \\ 0& \cos \varphi & -\sin \varphi \\ 0& \sin \varphi /\cos \theta & \cos \varphi /\cos \theta \end{array}\right]\left[\begin{array}{c}{\omega }_x\\ {\omega }_y\\ {\omega }_z\end{array}\right]$$