[足式机器人]Part3 机构运动学与动力学分析与建模 Ch00-5 刚体的加速度与角加速度

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
若有帮助请引用
本文参考：
.

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

5. 运动刚体的加速度与角加速度

5.1 矢量的速度与加速度

矢量的速度与加速度，不同于点的速度与加速度——描述该矢量在对应方向上的延长与收缩情况（模值的变大与减小）：
对于矢量的速度而言，有：
$$\begin{gathered} R_{\mathrm{Vector}}^F=\left[Q_{\mathrm{M}}^F\right]R_{\mathrm{Vector}}^M \\ \Rightarrow {\dot{R}}_{\mathrm{Vector}}^F=\left[{\dot{Q}}_{\mathrm{M}}^F\right]R_{\mathrm{Vector}}^M+\left[Q_{\mathrm{M}}^F\right]{\dot{R}}_{\mathrm{Vector}}^M=\left[Q_{\mathrm{M}}^F\right]{\dot{R}}_{\mathrm{Vector}}^M+{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]R_{\mathrm{Vector}}^M \end{gathered}$$
对于矢量的加速度而言，有：
$$\begin{gathered} {\dot{R}}_{\mathrm{Vector}}^F=\left[Q_{\mathrm{M}}^F\right]{\dot{R}}_{\mathrm{Vector}}^M+{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]R_{\mathrm{Vector}}^M \\ \Rightarrow {\ddot{R}}_{\mathrm{Vector}}^F=\left[{\dot{Q}}_{\mathrm{M}}^F\right]{\dot{R}}_{\mathrm{Vector}}^M+\left[Q_{\mathrm{M}}^F\right]{\ddot{R}}_{\mathrm{Vector}}^M+{\dot{\tilde{\omega }}}^F\left[Q_{\mathrm{M}}^F\right]R_{\mathrm{Vector}}^M+{\tilde{\omega }}^F\left[{\dot{Q}}_{\mathrm{M}}^F\right]R_{\mathrm{Vector}}^M+{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]{\dot{R}}_{\mathrm{Vector}}^M \\ \Rightarrow {\ddot{R}}_{\mathrm{Vector}}^F={\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]{\dot{R}}_{\mathrm{Vector}}^M+\left[Q_{\mathrm{M}}^F\right]{\ddot{R}}_{\mathrm{Vector}}^M+{\dot{\tilde{\omega }}}^F\left[Q_{\mathrm{M}}^F\right]R_{\mathrm{Vector}}^M+{\tilde{\omega }}^F{\dot{\tilde{\omega }}}^F\left[Q_{\mathrm{M}}^F\right]R_{\mathrm{Vector}}^M+{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]{\dot{R}}_{\mathrm{Vector}}^M \\ \Rightarrow {\ddot{R}}_{\mathrm{Vector}}^F=a_{\mathrm{Vector}}^F+2{\tilde{\omega }}^F{v}_{\mathrm{Vector}}^F+{\dot{\tilde{\omega }}}^F{R}_{\mathrm{Vector}}^F+{\tilde{\omega }}^F{\tilde{\omega }}^F{R}_{\mathrm{Vector}}^F \end{gathered}$$
整理可得：
$${\ddot{R}}_{\mathrm{Vector}}^F=a_{\mathrm{Vector}}^F+2{\tilde{\omega }}^F{v}_{\mathrm{Vector}}^F+{\tilde{\alpha }}^F{R}_{\mathrm{Vector}}^F+{\tilde{\omega }}^F{\tilde{\omega }}^F{R}_{\mathrm{Vector}}^F$$

其中：

• $a_{\mathrm{Vector}}^F=\left[Q_{\mathrm{M}}^F\right]{\ddot{R}}_{\mathrm{Vector}}^M$ 运动坐标系下的矢量加速度：表示坐标系 { F } \left\{ F \right\} {F}下定义的坐标系 { M } \left\{ M \right\} {M}的矢量加速度，若矢量的模固定（不发生变化），即在坐标系 { M } \left\{ M \right\} {M}中具有固定的投影参数，则该项为$0$。
• $2{\tilde{\omega }}^F{v}_{\mathrm{Vector}}^F=2{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]{\dot{R}}_{\mathrm{Vector}}^M$：科里奥利分量：其方向同时垂直于${\omega }^F$与$v_{\mathrm{Vector}}^F$
• ${\tilde{\alpha }}^F{R}_{\mathrm{Vector}}^F={\dot{\tilde{\omega }}}^F{R}_{\mathrm{Vector}}^F$切向分量：其方向同时垂直于${\alpha }^F$与$R_{\mathrm{Vector}}^F$
• ${\tilde{\omega }}^F{\tilde{\omega }}^F{R}_{\mathrm{Vector}}^F$法向分量：其方向为 $-R_{\mathrm{Vector}}^F$

5.2 点的速度与加速度

对 $v_{{\mathrm{P}}_{\mathrm{i}}}^M=\left({\tilde{\omega }}^M-{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]\right)R_{{\mathrm{P}}_{\mathrm{i}}}^M$ 进一步求导，可计算出其运动刚体上点$P_i$的加速度为：
$$v_{{\mathrm{P}}_{\mathrm{i}}}^F=v_{\mathrm{M}}^F+\left[{\dot{Q}}_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]{\dot{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M=\{\begin{array}{l}{v}_{\mathrm{M}}^F+{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]v_{{\mathrm{P}}_{\mathrm{i}}}^M\\ v_{\mathrm{M}}^F+\left[Q_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{R}_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]v_{{\mathrm{P}}_{\mathrm{i}}}^M\end{array}$$

• 坐标系 { F } \left\{ F \right\} {F}下的推导：
  $$\begin{array}{rl}{a}_{{\mathrm{P}}_{\mathrm{i}}}^F& =a_{\mathrm{M}}^F+\left({\dot{\tilde{\omega }}}^F\left[Q_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M+{\tilde{\omega }}^F\left[{\dot{Q}}_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M+{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]v_{{\mathrm{P}}_{\mathrm{i}}}^M\right)+\left(\left[{\dot{Q}}_{\mathrm{M}}^F\right]v_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]a_{{\mathrm{P}}_{\mathrm{i}}}^M\right)\\ \Rightarrow a_{{\mathrm{P}}_{\mathrm{i}}}^F& =a_{\mathrm{M}}^F+\left({\tilde{\alpha }}^F\left[Q_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M+{\tilde{\omega }}^F{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M+{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]v_{{\mathrm{P}}_{\mathrm{i}}}^M\right)+\left({\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]v_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]a_{{\mathrm{P}}_{\mathrm{i}}}^M\right)\\ \Rightarrow a_{{\mathrm{P}}_{\mathrm{i}}}^F& =a_{\mathrm{M}}^F+{\tilde{\alpha }}^F{\left(R_{{\mathrm{P}}_{\mathrm{i}}}^M\right)}^F+{\tilde{\omega }}^F{\tilde{\omega }}^F{\left(R_{{\mathrm{P}}_{\mathrm{i}}}^M\right)}^F+2{\tilde{\omega }}^F{\left(v_{{\mathrm{P}}_{\mathrm{i}}}^M\right)}^F+{\left(a_{{\mathrm{P}}_{\mathrm{i}}}^M\right)}^F\end{array}$$
  当点$P_i$为运动刚体上固定一点时，则有：
  $$\begin{array}{rl}{a}_{{\mathrm{P}}_{\mathrm{i}}}^F& =a_{\mathrm{M}}^F+{\tilde{\alpha }}^F{\left(R_{{\mathrm{P}}_{\mathrm{i}}}^M\right)}^F+{\tilde{\omega }}^F{\tilde{\omega }}^F{\left(R_{{\mathrm{P}}_{\mathrm{i}}}^M\right)}^F+2{\tilde{\omega }}^F{\left({v_{{\mathrm{P}}_{\mathrm{i}}}^M}_{\nearrow 0}\right)}^F+{\left({a_{{\mathrm{P}}_{\mathrm{i}}}^M}_{\nearrow 0}\right)}^F\\ & =a_{\mathrm{M}}^F+{\tilde{\alpha }}^F{\left(R_{{\mathrm{P}}_{\mathrm{i}}}^M\right)}^F+{\tilde{\omega }}^F{\tilde{\omega }}^F{\left(R_{{\mathrm{P}}_{\mathrm{i}}}^M\right)}^F\end{array}$$
• 坐标系 { M } \left\{ M \right\} {M}下的推导：
  $$\begin{array}{rl}{a}_{{\mathrm{P}}_{\mathrm{i}}}^F& =a_{\mathrm{M}}^F+\left(\left[{\dot{Q}}_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{R}_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]{\dot{\tilde{\omega }}}^M{R}_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{\dot{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M\right)+\left(\left[{\dot{Q}}_{\mathrm{M}}^F\right]v_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]a_{{\mathrm{P}}_{\mathrm{i}}}^M\right)\\ \Rightarrow a_{{\mathrm{P}}_{\mathrm{i}}}^F& =a_{\mathrm{M}}^F+\left(\left[Q_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{\tilde{\omega }}^M{R}_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]{\tilde{\alpha }}^M{R}_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{v}_{{\mathrm{P}}_{\mathrm{i}}}^M\right)+\left(\left[Q_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{v}_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]a_{{\mathrm{P}}_{\mathrm{i}}}^M\right)\\ \Rightarrow a_{{\mathrm{P}}_{\mathrm{i}}}^F& =a_{\mathrm{M}}^F+\left[Q_{\mathrm{M}}^F\right]{\tilde{\alpha }}^M{R}_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{\tilde{\omega }}^M{R}_{{\mathrm{P}}_{\mathrm{i}}}^M+2\left[Q_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{v}_{{\mathrm{P}}_{\mathrm{i}}}^M+{\left(a_{{\mathrm{P}}_{\mathrm{i}}}^M\right)}^F\end{array}$$
  当点$P_i$为运动刚体上固定一点时，则有：
  $$\begin{array}{rl}{a}_{{\mathrm{P}}_{\mathrm{i}}}^F& =a_{\mathrm{M}}^F+\left[Q_{\mathrm{M}}^F\right]{\tilde{\alpha }}^M{R}_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{\tilde{\omega }}^M{R}_{{\mathrm{P}}_{\mathrm{i}}}^M+2\left[Q_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{v_{{\mathrm{P}}_{\mathrm{i}}}^M}_{\nearrow 0}+{\left({a_{{\mathrm{P}}_{\mathrm{i}}}^M}_{\nearrow 0}\right)}^F\\ & =a_{\mathrm{M}}^F+\left[Q_{\mathrm{M}}^F\right]{\tilde{\alpha }}^M{R}_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{\tilde{\omega }}^M{R}_{{\mathrm{P}}_{\mathrm{i}}}^M\end{array}$$

5.2.1 欧拉角表示角加速度

对 ω ⃗ F = [ cos ⁡ β cos ⁡ γ − sin ⁡ γ 0 cos ⁡ β sin ⁡ γ cos ⁡ γ 0 − sin ⁡ β 0 1 ] [ α ˙ β ˙ γ ˙ ] \vec{\omega}^F=\left[ \begin{matrix} \cos \beta \cos \gamma& -\sin \gamma& 0\\ \cos \beta \sin \gamma& \cos \gamma& 0\\ -\sin \beta& 0& 1\\ \end{matrix} \right] \left[ \begin{array}{c} \dot{\alpha}\\ \dot{\beta}\\ \dot{\gamma}\\ \end{array} \right] ω F= ​cosβcosγcosβsinγ−sinβ​−sinγcosγ0​001​ ​ ​α˙β˙​γ˙​​ ​ 继续求导，可得：
ω ⃗ F = [ cos ⁡ β cos ⁡ γ − sin ⁡ γ 0 cos ⁡ β sin ⁡ γ cos ⁡ γ 0 − sin ⁡ β 0 1 ] [ α ˙ β ˙ γ ˙ ] ⇒ α ⃗ F = [ − sin ⁡ β cos ⁡ γ − cos ⁡ β sin ⁡ γ − cos ⁡ γ 0 cos ⁡ β cos ⁡ γ − sin ⁡ β sin ⁡ γ − sin ⁡ γ 0 − cos ⁡ β 0 0 ] [ α ˙ β ˙ γ ˙ ] + [ cos ⁡ β cos ⁡ γ − sin ⁡ γ 0 cos ⁡ β sin ⁡ γ cos ⁡ γ 0 − sin ⁡ β 0 1 ] [ α ¨ β ¨ γ ¨ ] \begin{split} \vec{\omega}^F=\left[ \begin{matrix} \cos \beta \cos \gamma& -\sin \gamma& 0\\ \cos \beta \sin \gamma& \cos \gamma& 0\\ -\sin \beta& 0& 1\\ \end{matrix} \right] \left[ \begin{array}{c} \dot{\alpha}\\ \dot{\beta}\\ \dot{\gamma}\\ \end{array} \right] \\ \Rightarrow \vec{\alpha}^F=\left[ \begin{matrix} -\sin \beta \cos \gamma -\cos \beta \sin \gamma& -\cos \gamma& 0\\ \cos \beta \cos \gamma -\sin \beta \sin \gamma& -\sin \gamma& 0\\ -\cos \beta& 0& 0\\ \end{matrix} \right] \left[ \begin{array}{c} \dot{\alpha}\\ \dot{\beta}\\ \dot{\gamma}\\ \end{array} \right] +\left[ \begin{matrix} \cos \beta \cos \gamma& -\sin \gamma& 0\\ \cos \beta \sin \gamma& \cos \gamma& 0\\ -\sin \beta& 0& 1\\ \end{matrix} \right] \left[ \begin{array}{c} \ddot{\alpha}\\ \ddot{\beta}\\ \ddot{\gamma}\\ \end{array} \right] \end{split} ω F= ​cosβcosγcosβsinγ−sinβ​−sinγcosγ0​001​ ​ ​α˙β˙​γ˙​​ ​⇒α F= ​−sinβcosγ−cosβsinγcosβcosγ−sinβsinγ−cosβ​−cosγ−sinγ0​000​ ​ ​α˙β˙​γ˙​​ ​+ ​cosβcosγcosβsinγ−sinβ​−sinγcosγ0​001​ ​ ​α¨β¨​γ¨​​ ​​

5.2.2 欧拉参数表示角加速度

结合欧拉参数对角速度的表达： [ w 1 F w 2 F w 3 F ] = 2 [ q ˙ 4 q 3 − q ˙ 3 q 4 + q ˙ 2 q 1 − q ˙ 1 q 2 q ˙ 2 q 4 − q ˙ 1 q 3 + q ˙ 4 q 2 − q ˙ 3 q 1 q ˙ 3 q 2 − q ˙ 4 q 1 + q ˙ 1 q 4 − q ˙ 2 q 3 ] \left[ \begin{array}{c} {w_1}^F\\ {w_2}^F\\ {w_3}^F\\ \end{array} \right] =2\left[ \begin{array}{c} \dot{q}_4q_3-\dot{q}_3q_4+\dot{q}_2q_1-\dot{q}_1q_2\\ \dot{q}_2q_4-\dot{q}_1q_3+\dot{q}_4q_2-\dot{q}_3q_1\\ \dot{q}_3q_2-\dot{q}_4q_1+\dot{q}_1q_4-\dot{q}_2q_3\\ \end{array} \right] ​w1​Fw2​Fw3​F​ ​=2 ​q˙​4​q3​−q˙​3​q4​+q˙​2​q1​−q˙​1​q2​q˙​2​q4​−q˙​1​q3​+q˙​4​q2​−q˙​3​q1​q˙​3​q2​−q˙​4​q1​+q˙​1​q4​−q˙​2​q3​​ ​对上式进一步求导，可得：
[ α 1 F α 2 F α 3 F ] = 2 [ q ¨ 4 q 3 − q ¨ 3 q 4 + q ¨ 2 q 1 − q ¨ 1 q 2 q ¨ 2 q 4 − q ¨ 1 q 3 + q ¨ 4 q 2 − q ¨ 3 q 1 q ¨ 3 q 2 − q ¨ 4 q 1 + q ¨ 1 q 4 − q ¨ 2 q 3 ] \left[ \begin{array}{c} {\alpha _1}^F\\ {\alpha _2}^F\\ {\alpha _3}^F\\ \end{array} \right] =2\left[ \begin{array}{c} \ddot{q}_4q_3-\ddot{q}_3q_4+\ddot{q}_2q_1-\ddot{q}_1q_2\\ \ddot{q}_2q_4-\ddot{q}_1q_3+\ddot{q}_4q_2-\ddot{q}_3q_1\\ \ddot{q}_3q_2-\ddot{q}_4q_1+\ddot{q}_1q_4-\ddot{q}_2q_3\\ \end{array} \right] ​α1​Fα2​Fα3​F​ ​=2 ​q¨​4​q3​−q¨​3​q4​+q¨​2​q1​−q¨​1​q2​q¨​2​q4​−q¨​1​q3​+q¨​4​q2​−q¨​3​q1​q¨​3​q2​−q¨​4​q1​+q¨​1​q4​−q¨​2​q3​​ ​
简化为：${\alpha }^F=2B{\ddot{q}}^F$，其中： B 3 × 4 = [ − q 2 q 1 − q 4 q 3 − q 3 q 4 q 1 − q 2 − q 4 − q 3 q 2 q 1 ] 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] B3×4​= ​−q2​−q3​−q4​​q1​q4​−q3​​−q4​q1​q2​​q3​−q2​q1​​ ​

5.3 欧拉参数的重要性质

根据前述定义： q ⃗ = [ s v ⃗ ] = [ cos ⁡ θ 2 → s c a l e    p a r t v ⃗ sin ⁡ θ 2 → v e c t o r    p a r t ] = [ cos ⁡ θ 2 v 1 sin ⁡ θ 2 v 2 sin ⁡ θ 2 v 3 sin ⁡ θ 2 ] = [ q 1 q 2 q 3 q 4 ] \vec{q}=\left[ \begin{array}{c} s\\ \vec{v}\\ \end{array} \right] =\left[ \begin{matrix} \cos \frac{\theta}{2}& \rightarrow scale\,\,part\\ \vec{v}\sin \frac{\theta}{2}& \rightarrow vector\,\,part\\ \end{matrix} \right] =\left[ \begin{array}{c} \cos \frac{\theta}{2}\\ v_1\sin \frac{\theta}{2}\\ v_2\sin \frac{\theta}{2}\\ v_3\sin \frac{\theta}{2}\\ \end{array} \right] =\left[ \begin{array}{c} q_1\\ q_2\\ q_3\\ q_4\\ \end{array} \right] q ​=[sv ​]=[cos2θ​v sin2θ​​→scalepart→vectorpart​]= ​cos2θ​v1​sin2θ​v2​sin2θ​v3​sin2θ​​ ​= ​q1​q2​q3​q4​​ ​， B 3 × 4 = [ − q 2 q 1 − q 4 q 3 − q 3 q 4 q 1 − q 2 − q 4 − q 3 q 2 q 1 ] 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] B3×4​= ​−q2​−q3​−q4​​q1​q4​−q3​​−q4​q1​q2​​q3​−q2​q1​​ ​， B ˉ 3 × 4 = [ − q 2 q 1 q 4 − q 3 − q 3 − q 4 q 1 q 2 − q 4 q 3 − q 2 q 1 ] \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] Bˉ3×4​= ​−q2​−q3​−q4​​q1​−q4​q3​​q4​q1​−q2​​−q3​q2​q1​​ ​, 总结其一些有用的性质，进而便于简化运动学与动力学方程：

$$\{\begin{array}{l}B{B}^{\mathrm{T}}=\bar{B}{\bar{B}}^{\mathrm{T}}=E\\ B^{\mathrm{T}}B={\bar{B}}^{\mathrm{T}}\bar{B}=E_{4\times 4}-q{q}^{\mathrm{T}}\end{array}$$

$$\{\begin{array}{l}Bq={\bar{B}}^{\mathrm{T}}q=0\\ \dot{B}q={\dot{\bar{B}}}^{\mathrm{T}}q=0\\ B{\dot{\bar{B}}}^{\mathrm{T}}=\dot{B}{\bar{B}}^{\mathrm{T}}\end{array}$$

$$\{\begin{array}{l}2\dot{B}{B}^{\mathrm{T}}=-2B{\dot{B}}^{\mathrm{T}}={\tilde{\omega }}^F\\ 2\bar{B}{\dot{\bar{B}}}^{\mathrm{T}}=-2\dot{\bar{B}}{\bar{B}}^{\mathrm{T}}={\tilde{\omega }}^M\end{array}$$

$$\{\begin{array}{l}B\dot{q}=-\dot{B}q\\ \bar{B}\dot{q}=-\dot{\bar{B}}q\end{array}$$

$$\{\begin{array}{l}{q}^{\mathrm{T}}q=1\\ {\dot{q}}^{\mathrm{T}}q=0\end{array}$$

考虑存在如下的旋转变换关系 : $\left[Q_{\mathrm{M}}^F\right]=\left[Q_{\mathrm{A}}^F\right]\left[Q_{\mathrm{M}}^A\right]$ , 各旋转矩阵对应的欧拉参数分别为：$q_{\to \mathrm{M}}^F,q_{\to \mathrm{A}}^F,q_{\to \mathrm{M}}^A$，则存在如下转换关系：
q ⃗ → M F = [ q 1 F → M q 2 F → M q 3 F → M q 4 F → M ] = D q ⃗ → M A = [ − q 1 F → A q 2 F → A q 3 F → A q 4 F → A − q 2 F → A − q 1 F → A q 4 F → A − q 3 F → A − q 3 F → A − q 4 F → A − q 1 F → A q 2 F → A − q 4 F → A q 3 F → A − q 2 F → A − q 1 F → A ] [ q 1 A → M q 2 A → M q 3 A → M q 4 A → M ] , D = [ − q 1 F → A q 2 F → A q 3 F → A q 4 F → A − q 2 F → A − q 1 F → A q 4 F → A − q 3 F → A − q 3 F → A − q 4 F → A − q 1 F → A q 2 F → A − q 4 F → A q 3 F → A − q 2 F → A − q 1 F → A ] \vec{q}_{\rightarrow \mathrm{M}}^{F}=\left[ \begin{array}{c} q_{1}^{F\rightarrow M}\\ q_{2}^{F\rightarrow M}\\ q_{3}^{F\rightarrow M}\\ q_{4}^{F\rightarrow M}\\ \end{array} \right] =D\vec{q}_{\rightarrow \mathrm{M}}^{A}=\left[ \begin{matrix} -q_{1}^{F\rightarrow A}& q_{2}^{F\rightarrow A}& q_{3}^{F\rightarrow A}& q_{4}^{F\rightarrow A}\\ -q_{2}^{F\rightarrow A}& -q_{1}^{F\rightarrow A}& q_{4}^{F\rightarrow A}& -q_{3}^{F\rightarrow A}\\ -q_{3}^{F\rightarrow A}& -q_{4}^{F\rightarrow A}& -q_{1}^{F\rightarrow A}& q_{2}^{F\rightarrow A}\\ -q_{4}^{F\rightarrow A}& q_{3}^{F\rightarrow A}& -q_{2}^{F\rightarrow A}& -q_{1}^{F\rightarrow A}\\ \end{matrix} \right] \left[ \begin{array}{c} q_{1}^{A\rightarrow M}\\ q_{2}^{A\rightarrow M}\\ q_{3}^{A\rightarrow M}\\ q_{4}^{A\rightarrow M}\\ \end{array} \right] ,D=\left[ \begin{matrix} -q_{1}^{F\rightarrow A}& q_{2}^{F\rightarrow A}& q_{3}^{F\rightarrow A}& q_{4}^{F\rightarrow A}\\ -q_{2}^{F\rightarrow A}& -q_{1}^{F\rightarrow A}& q_{4}^{F\rightarrow A}& -q_{3}^{F\rightarrow A}\\ -q_{3}^{F\rightarrow A}& -q_{4}^{F\rightarrow A}& -q_{1}^{F\rightarrow A}& q_{2}^{F\rightarrow A}\\ -q_{4}^{F\rightarrow A}& q_{3}^{F\rightarrow A}& -q_{2}^{F\rightarrow A}& -q_{1}^{F\rightarrow A}\\ \end{matrix} \right] q ​→MF​= ​q1F→M​q2F→M​q3F→M​q4F→M​​ ​=Dq ​→MA​= ​−q1F→A​−q2F→A​−q3F→A​−q4F→A​​q2F→A​−q1F→A​−q4F→A​q3F→A​​q3F→A​q4F→A​−q1F→A​−q2F→A​​q4F→A​−q3F→A​q2F→A​−q1F→A​​ ​ ​q1A→M​q2A→M​q3A→M​q4A→M​​ ​,D= ​−q1F→A​−q2F→A​−q3F→A​−q4F→A​​q2F→A​−q1F→A​−q4F→A​q3F→A​​q3F→A​q4F→A​−q1F→A​−q2F→A​​q4F→A​−q3F→A​q2F→A​−q1F→A​​ ​文章来源地址https://www.toymoban.com/news/detail-823176.html