本文仅供学习使用，总结很多本现有讲述运动学或动力学书籍后的总结，从矢量的角度进行分析，方法比较传统，但更易理解，并且现有的看似抽象方法，两者本质上并无不同。

2024年底本人学位论文发表后方可摘抄
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本文参考：
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食用方法
求解逻辑：速度与加速度都是在知道角速度与角加速度的前提下——旋转运动更重要
所求得的速度表达-需要考虑是否为刚体相对固定点！
旋转矩阵？转换矩阵？有什么意义和性质？——与角速度与角加速度的关系
务必自己推导全部公式，并理解每个符号的含义

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4. 刚体的速度与角速度

对于运动坐标系下任意一点$P_{\mathrm{i}}$而言，有：
$$\begin{array}{rl}& {\vec{R}}_{\mathrm{P}}^F={\vec{R}}_{\mathrm{M}}^F+\left[Q_{\mathrm{M}}^F\right]{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M\\ & \Rightarrow {\vec{v}}_{\mathrm{P}}^F={\vec{v}}_{\mathrm{M}}^F+\left[{\dot{Q}}_{\mathrm{M}}^F\right]{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]{\dot{\vec{R}}}_{{\mathrm{P}}_{\mathrm{i}}}^M={\vec{\omega }}^F\times {\vec{R}}_{\mathrm{P}}^F={\widetilde{\stackrel{⃗}{\omega }}}^F{\vec{R}}_{\mathrm{P}}^F={\widetilde{\stackrel{⃗}{\omega }}}^F\left({\vec{R}}_{\mathrm{M}}^F+\left[Q_{\mathrm{M}}^F\right]{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M\right)\\ & \Rightarrow \left[{\dot{Q}}_{\mathrm{M}}^F\right]{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]{\dot{\vec{R}}}_{{\mathrm{P}}_{\mathrm{i}}}^M={\widetilde{\stackrel{⃗}{\omega }}}^F\left[Q_{\mathrm{M}}^F\right]{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M\\ & \Rightarrow {\vec{v}}_{{\mathrm{P}}_{\mathrm{i}}}^M=\left({\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}{\widetilde{\stackrel{⃗}{\omega }}}^F\left[Q_{\mathrm{M}}^F\right]-{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]\right){\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M=\left(\tilde{\left({\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}{\vec{\omega }}^F\right)}-{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]\right){\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M=\left({\widetilde{\stackrel{⃗}{\omega }}}^M-{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]\right){\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M\end{array}$$

因此，当$P_{\mathrm{i}}$为刚体上的固定点时，有：${\vec{v}}_{{\mathrm{P}}_{\mathrm{i}}}^M=0$，进而可知：
$$\begin{array}{rl}{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}{\widetilde{\stackrel{⃗}{\omega }}}^F\left[Q_{\mathrm{M}}^F\right]-{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]=0& \Rightarrow {\widetilde{\stackrel{⃗}{\omega }}}^F=\left[{\dot{Q}}_{\mathrm{M}}^F\right]{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\\ {\widetilde{\stackrel{⃗}{\omega }}}^M-{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]=0& \Rightarrow {\widetilde{\stackrel{⃗}{\omega }}}^M={\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]\end{array}$$

对转换矩阵${\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}$而言，有：
$$\begin{array}{rl}& {\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[Q_{\mathrm{M}}^F\right]=0\\ \Rightarrow & {\left[{\dot{Q}}_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[Q_{\mathrm{M}}^F\right]+{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]=0\\ \Rightarrow & {\left[{\dot{Q}}_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[Q_{\mathrm{M}}^F\right]+{\left[{\left[{\dot{Q}}_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[Q_{\mathrm{M}}^F\right]\right]}^{\mathrm{T}}=0\end{array}$$
即，${\left[{\dot{Q}}_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[Q_{\mathrm{M}}^F\right]$为反(斜)对称矩阵。

因此，对于矩阵 ${\widetilde{\stackrel{⃗}{\omega }}}^F$与 ${\widetilde{\stackrel{⃗}{\omega }}}^M$具有如下转换关系：
$$\begin{array}{rl}{\widetilde{\stackrel{⃗}{\omega }}}^M& ={\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}{\widetilde{\stackrel{⃗}{\omega }}}^F\left[Q_{\mathrm{M}}^F\right]\\ {\widetilde{\stackrel{⃗}{\omega }}}^F& =\left[Q_{\mathrm{M}}^F\right]{\widetilde{\stackrel{⃗}{\omega }}}^M{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\end{array}$$

进而可将上式中的项term $\left[{\dot{Q}}_{\mathrm{M}}^F\right]{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M$改写为（下式仅当$P_{\mathrm{i}}$为刚体上的固定点时成立）：
$$\left[{\dot{Q}}_{\mathrm{M}}^F\right]{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M=\{\begin{array}{l}{\widetilde{\stackrel{⃗}{\omega }}}^F\left[Q_{\mathrm{M}}^F\right]{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M={\widetilde{\stackrel{⃗}{\omega }}}^F{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^F={\vec{\omega }}^F\times {\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^F\\ \left[Q_{\mathrm{M}}^F\right]{\widetilde{\stackrel{⃗}{\omega }}}^M{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M=\left[Q_{\mathrm{M}}^F\right]\left({\vec{\omega }}^M\times {\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M\right)\end{array}$$

4.1 角速度的表达

4.1.1 欧拉参数表示角速度

结合定义矩阵：$B_{3\times 4}=\left[\begin{array}{cccc}-q_2& q_1& -q_4& q_3\\ -q_3& q_4& q_1& -q_2\\ -q_4& -q_3& q_2& q_1\end{array}\right]$${\bar{B}}_{3\times 4}=\left[\begin{array}{cccc}-q_2& q_1& q_4& -q_3\\ -q_3& -q_4& q_1& q_2\\ -q_4& q_3& -q_2& q_1\end{array}\right]$, 带入同样的式子可得：

$$\begin{array}{rl}{\widetilde{\stackrel{⃗}{\omega }}}^F& =2\bar{B}{\dot{\bar{B}}}^{\mathrm{T}}\\ {\widetilde{\stackrel{⃗}{\omega }}}^M& =2B{\dot{B}}^{\mathrm{T}}\end{array}$$
将上式展开，由四元数的归一化可知：${\dot{q}}_1{q}_1+{\dot{q}}_2{q}_2+{\dot{q}}_3{q}_3+{\dot{q}}_4{q}_4=0$，可得：
$$\begin{array}{rl}\left[\begin{array}{c}{w_1}^F\\ {w_2}^F\\ {w_3}^F\end{array}\right]& =2\left[\begin{array}{c}{\dot{q}}_4{q}_3-{\dot{q}}_3{q}_4+{\dot{q}}_2{q}_1-{\dot{q}}_1{q}_2\\ {\dot{q}}_2{q}_4-{\dot{q}}_1{q}_3+{\dot{q}}_4{q}_2-{\dot{q}}_3{q}_1\\ {\dot{q}}_3{q}_2-{\dot{q}}_4{q}_1+{\dot{q}}_1{q}_4-{\dot{q}}_2{q}_3\end{array}\right]\\ \left[\begin{array}{c}{w_1}^M\\ {w_2}^M\\ {w_3}^M\end{array}\right]& =2\left[\begin{array}{c}{q}_4{\dot{q}}_3-q_3{\dot{q}}_4-q_2{\dot{q}}_1+q_1{\dot{q}}_2\\ q_2{\dot{q}}_4+q_1{\dot{q}}_3-q_4{\dot{q}}_2-q_3{\dot{q}}_1\\ q_3{\dot{q}}_2-q_4{\dot{q}}_1+q_1{\dot{q}}_4-q_2{\dot{q}}_3\end{array}\right]\end{array}$$
继续观察上式，将上式进行化简：
$$\begin{gathered} {\vec{\omega }}^F=2B\dot{\vec{q}}=-2\dot{B}\vec{q} \\ {\vec{\omega }}^M=2\bar{B}\dot{\vec{q}}=-2\dot{\bar{B}}\vec{q} \end{gathered}$$

进而可将$\left[{\dot{Q}}_{\mathrm{M}}^F\right]{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M=\left[Q_{\mathrm{M}}^F\right]\left({\vec{\omega }}^M\times {\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M\right)$改写为（下式仅当$P_{\mathrm{i}}$为刚体上的固定点时成立）：
$$\left[{\dot{Q}}_{\mathrm{M}}^F\right]{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M=\left[Q_{\mathrm{M}}^F\right]\left({\vec{\omega }}^M\times {\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M\right)=-\left[Q_{\mathrm{M}}^F\right]\left({\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M\times {\vec{\omega }}^M\right)=-\left[Q_{\mathrm{M}}^F\right]{\widetilde{\stackrel{⃗}{R}}}_{{\mathrm{P}}_{\mathrm{i}}}^M\left(2\bar{B}\dot{\vec{q}}\right)$$

因为所有表达方式都能转换成欧拉参数-四元数的形式，因此上式在计算过程中具有普适性。

进而可知：
$$\frac{\partial \left(\left[Q_{\mathrm{M}}^F\right]{\vec{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M\right)}{\partial \vec{q}}=-\left[Q_{\mathrm{M}}^F\right]{\widetilde{\stackrel{⃗}{R}}}_{{\mathrm{P}}_{\mathrm{i}}}^M\left(2\bar{B}\right)$$

4.1.2 轴角参数表示角速度

将$\left[\begin{array}{c}\theta \\ v_1\\ v_2\\ v_3\end{array}\right]=\left[\begin{array}{c}2\mathrm{arc}\cos \left(q_1\right)\\ \frac{q_2}{\sin \frac{\theta }{2}}\\ \frac{q_3}{\sin \frac{\theta }{2}}\\ \frac{q_4}{\sin \frac{\theta }{2}}\end{array}\right]$带入$\left[\begin{array}{c}{w_1}^F\\ {w_2}^F\\ {w_3}^F\end{array}\right]=2\left[\begin{array}{c}{\dot{q}}_4{q}_3-{\dot{q}}_3{q}_4+{\dot{q}}_2{q}_1-{\dot{q}}_1{q}_2\\ {\dot{q}}_2{q}_4-{\dot{q}}_1{q}_3+{\dot{q}}_4{q}_2-{\dot{q}}_3{q}_1\\ {\dot{q}}_3{q}_2-{\dot{q}}_4{q}_1+{\dot{q}}_1{q}_4-{\dot{q}}_2{q}_3\end{array}\right],\left[\begin{array}{c}{w_1}^M\\ {w_2}^M\\ {w_3}^M\end{array}\right]=2\left[\begin{array}{c}{q}_4{\dot{q}}_3-q_3{\dot{q}}_4-q_2{\dot{q}}_1+q_1{\dot{q}}_2\\ q_2{\dot{q}}_4+q_1{\dot{q}}_3-q_4{\dot{q}}_2-q_3{\dot{q}}_1\\ q_3{\dot{q}}_2-q_4{\dot{q}}_1+q_1{\dot{q}}_4-q_2{\dot{q}}_3\end{array}\right]$可得：
$$\begin{array}{rl}\left[\begin{array}{c}{w_1}^F\\ {w_2}^F\\ {w_3}^F\end{array}\right]& =\left[\begin{array}{c}2\left({\dot{v}}_3{v}_2-{\dot{v}}_2{v}_3\right){\sin }^2\frac{\theta }{2}+{\dot{v}}_1\sin \theta +\dot{\theta }{v}_1\\ 2\left({\dot{v}}_1{v}_3-{\dot{v}}_3{v}_1\right){\sin }^2\frac{\theta }{2}+{\dot{v}}_2\sin \theta +\dot{\theta }{v}_2\\ 2\left({\dot{v}}_2{v}_1-{\dot{v}}_1{v}_2\right){\sin }^2\frac{\theta }{2}+{\dot{v}}_3\sin \theta +\dot{\theta }{v}_3\end{array}\right]\\ \left[\begin{array}{c}{w_1}^M\\ {w_2}^M\\ {w_3}^M\end{array}\right]& =\left[\begin{array}{c}2\left(v_3{\dot{v}}_2-v_2{\dot{v}}_3\right){\sin }^2\frac{\theta }{2}+{\dot{v}}_1\sin \theta +\dot{\theta }{v}_1\\ 2\left(v_1{\dot{v}}_3-v_3{\dot{v}}_1\right){\sin }^2\frac{\theta }{2}+{\dot{v}}_2\sin \theta +\dot{\theta }{v}_2\\ 2\left(v_2{\dot{v}}_1-v_1{\dot{v}}_2\right){\sin }^2\frac{\theta }{2}+{\dot{v}}_3\sin \theta +\dot{\theta }{v}_3\end{array}\right]\end{array}$$

整理为：
$$\begin{array}{rl}{\vec{\omega }}^F& =2{\vec{v}}^F\times {\dot{\vec{v}}}^F{\sin }^2\frac{\theta }{2}+{\dot{\vec{v}}}^F\sin \theta +\dot{\theta }{\vec{v}}^F\\ {\vec{\omega }}^M& =2{\dot{\vec{v}}}^F\times {\vec{v}}^F{\sin }^2\frac{\theta }{2}+{\dot{\vec{v}}}^F\sin \theta +\dot{\theta }{\vec{v}}^F\end{array}$$

4.1.3 轴角参数表示角速度

对于ZYX欧拉角而言，有：
$$\begin{array}{rl}& \{\begin{array}{l}\left[Q_{\mathrm{M}}^F\right]=\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]\left[Q_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]\left[Q_{{\mathrm{F}}_1}^F\left({\vec{i}}^F,\alpha \right)\right]\\ {\widetilde{\stackrel{⃗}{\omega }}}^F=\left[{\dot{Q}}_{\mathrm{M}}^F\right]{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\end{array}\\ {\widetilde{\stackrel{⃗}{\omega }}}^F& =\{\begin{array}{l}\left[{\dot{Q}}_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]\left[Q_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]\left[Q_{{\mathrm{F}}_1}^F\left({\vec{i}}^F,\alpha \right)\right]\cdot {\left[Q_{{\mathrm{F}}_1}^F\left({\vec{i}}^F,\alpha \right)\right]}^{\mathrm{T}}{\left[Q_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]}^{\mathrm{T}}{\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]}^{\mathrm{T}}+\\ \left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]\left[{\dot{Q}}_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]\left[Q_{{\mathrm{F}}_1}^F\left({\vec{i}}^F,\alpha \right)\right]\cdot {\left[Q_{{\mathrm{F}}_1}^F\left({\vec{i}}^F,\alpha \right)\right]}^{\mathrm{T}}{\left[Q_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]}^{\mathrm{T}}{\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]}^{\mathrm{T}}+\\ \left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]\left[Q_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]\left[{\dot{Q}}_{{\mathrm{F}}_1}^F\left({\vec{i}}^F,\alpha \right)\right]\cdot {\left[Q_{{\mathrm{F}}_1}^F\left({\vec{i}}^F,\alpha \right)\right]}^{\mathrm{T}}{\left[Q_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]}^{\mathrm{T}}{\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]}^{\mathrm{T}}\end{array}\\ {\widetilde{\stackrel{⃗}{\omega }}}^F& =\{\begin{array}{l}\left[{\dot{Q}}_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]{\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]}^{\mathrm{T}}+\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]\left[{\dot{Q}}_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]{\left[Q_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]}^{\mathrm{T}}{\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]}^{\mathrm{T}}\\ +\left[\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]\left[Q_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]\right]\left[{\dot{Q}}_{{\mathrm{F}}_1}^F\left({\vec{i}}^F,\alpha \right)\right]\cdot {\left[Q_{{\mathrm{F}}_1}^F\left({\vec{i}}^F,\alpha \right)\right]}^{\mathrm{T}}{\left[\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]\left[Q_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]\right]}^{\mathrm{T}}\end{array}\\ {\widetilde{\stackrel{⃗}{\omega }}}^F& ={\widetilde{\stackrel{⃗}{\omega }}}_{{\mathrm{F}}_3\left(M\right)}^{F_2}+\tilde{\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]{\widetilde{\stackrel{⃗}{\omega }}}_{{\mathrm{F}}_2}^{F_1}}+\tilde{\left[\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]\left[Q_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]\right]{\widetilde{\stackrel{⃗}{\omega }}}_{{\mathrm{F}}_1}^F}\\ \Rightarrow {\vec{\omega }}^F& ={\vec{\omega }}_{{\mathrm{F}}_3\left(M\right)}^{F_2}+\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]{\vec{\omega }}_{{\mathrm{F}}_2}^{F_1}+\left[\left[Q_{{\mathrm{F}}_3\left(M\right)}^{F_2}\left({\vec{k}}^F,\gamma \right)\right]\left[Q_{{\mathrm{F}}_2}^{F_1}\left({\vec{j}}^F,\beta \right)\right]\right]{\vec{\omega }}_{{\mathrm{F}}_1}^F\\ \Rightarrow {\vec{\omega }}^F& =\left[\begin{array}{c}0\\ 0\\ \dot{\gamma }\end{array}\right]+\left[\begin{array}{ccc}\cos \gamma & -\sin \gamma & 0\\ \sin \gamma & \cos \gamma & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}0\\ \dot{\beta }\\ 0\end{array}\right]+\left[\begin{array}{ccc}\cos \gamma & -\sin \gamma & 0\\ \sin \gamma & \cos \gamma & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \beta & 0& \sin \beta \\ 0& 1& 0\\ -\sin \beta & 0& \cos \beta \end{array}\right]\left[\begin{array}{c}\dot{\alpha }\\ 0\\ 0\end{array}\right]\\ \Rightarrow {\vec{\omega }}^F& =\left[\begin{array}{ccc}\cos \beta \cos \gamma & -\sin \gamma & 0\\ \cos \beta \sin \gamma & \cos \gamma & 0\\ -\sin \beta & 0& 1\end{array}\right]\left[\begin{array}{c}\dot{\alpha }\\ \dot{\beta }\\ \dot{\gamma }\end{array}\right]\end{array}$$