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
机构运动学与动力学分析与建模 Ch00-4 刚体的速度与角速度 Part1
- [4. 刚体的速度与角速度](#4. 刚体的速度与角速度)
-
- [4.1 角速度的表达](#4.1 角速度的表达)
-
- [4.1.1 欧拉参数表示角速度](#4.1.1 欧拉参数表示角速度)
- [4.1.2 轴角参数表示角速度](#4.1.2 轴角参数表示角速度)
- [4.1.3 轴角参数表示角速度](#4.1.3 轴角参数表示角速度)
4. 刚体的速度与角速度
对于运动坐标系下任意一点 P i P_{\mathrm{i}} Pi而言，有：
R ⃗ P F = R ⃗ M F + [ Q M F ] R ⃗ P i M ⇒ v ⃗ P F = v ⃗ M F + [ Q ˙ M F ] R ⃗ P i M + [ Q M F ] R ⃗ ˙ P i M = ω ⃗ F × R ⃗ P F = ω ⃗ ~ F R ⃗ P F = ω ⃗ ~ F ( R ⃗ M F + [ Q M F ] R ⃗ P i M ) ⇒ [ Q ˙ M F ] R ⃗ P i M + [ Q M F ] R ⃗ ˙ P i M = ω ⃗ ~ F [ Q M F ] R ⃗ P i M ⇒ v ⃗ P i M = ( [ Q M F ] T ω ⃗ ~ F [ Q M F ] − [ Q M F ] T [ Q ˙ M F ] ) R ⃗ P i M = ( ( [ Q M F ] T ω ⃗ F ) ~ − [ Q M F ] T [ Q ˙ M F ] ) R ⃗ P i M = ( ω ⃗ ~ M − [ Q M F ] T [ Q ˙ M F ] ) R ⃗ P i M \begin{split} &\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}=\tilde{\vec{\omega}}^F\vec{R}{\mathrm{P}}^{F}=\tilde{\vec{\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}=\tilde{\vec{\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}}\tilde{\vec{\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( \widetilde{\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( \tilde{\vec{\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{split} R PF=R MF+[QMF]R PiM⇒v PF=v MF+[Q˙MF]R PiM+[QMF]R ˙PiM=ω F×R PF=ω ~FR PF=ω ~F(R MF+[QMF]R PiM)⇒[Q˙MF]R PiM+[QMF]R ˙PiM=ω ~F[QMF]R PiM⇒v PiM=([QMF]Tω ~F[QMF]−[QMF]T[Q˙MF])R PiM=(([QMF]Tω F) −[QMF]T[Q˙MF])R PiM=(ω ~M−[QMF]T[Q˙MF])R PiM
因此，当 P i P_{\mathrm{i}} Pi为刚体上的固定点时，有： v ⃗ P i M = 0 \vec{v}{\mathrm{P}{\mathrm{i}}}^{M}=0 v PiM=0，进而可知：
[ Q M F ] T ω ⃗ ~ F [ Q M F ] − [ Q M F ] T [ Q ˙ M F ] = 0 ⇒ ω ⃗ ~ F = [ Q ˙ M F ] [ Q M F ] T ω ⃗ ~ M − [ Q M F ] T [ Q ˙ M F ] = 0 ⇒ ω ⃗ ~ M = [ Q M F ] T [ Q ˙ M F ] \begin{split} \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}}\tilde{\vec{\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 \tilde{\vec{\omega}}^F=\left[ \dot{Q}{\mathrm{M}}^{F} \right] \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}} \\ \tilde{\vec{\omega}}^M-\left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}}\left[ \dot{Q}{\mathrm{M}}^{F} \right] =0&\Rightarrow \tilde{\vec{\omega}}^M=\left[ Q{\mathrm{M}}^{F} \right] ^{\mathrm{T}}\left[ \dot{Q}_{\mathrm{M}}^{F} \right] \end{split} [QMF]Tω ~F[QMF]−[QMF]T[Q˙MF]=0ω ~M−[QMF]T[Q˙MF]=0⇒ω ~F=[Q˙MF][QMF]T⇒ω ~M=[QMF]T[Q˙MF]
对转换矩阵 [ Q M F ] T \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}} [QMF]T而言，有：
[ Q M F ] T [ Q M F ] = 0 ⇒ [ Q ˙ M F ] T [ Q M F ] + [ Q M F ] T [ Q ˙ M F ] = 0 ⇒ [ Q ˙ M F ] T [ Q M F ] + [ [ Q ˙ M F ] T [ Q M F ] ] T = 0 \begin{split} &\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{split} ⇒⇒[QMF]T[QMF]=0[Q˙MF]T[QMF]+[QMF]T[Q˙MF]=0[Q˙MF]T[QMF]+[[Q˙MF]T[QMF]]T=0
即， [ Q ˙ M F ] T [ Q M F ] \left[ \dot{Q}{\mathrm{M}}^{F} \right] ^{\mathrm{T}}\left[ Q{\mathrm{M}}^{F} \right] [Q˙MF]T[QMF]为反(斜)对称矩阵。
因此，对于矩阵 ω ⃗ ~ F \tilde{\vec{\omega}}^F ω ~F与 ω ⃗ ~ M \tilde{\vec{\omega}}^M ω ~M具有如下转换关系：
ω ⃗ ~ M = [ Q M F ] T ω ⃗ ~ F [ Q M F ] ω ⃗ ~ F = [ Q M F ] ω ⃗ ~ M [ Q M F ] T \begin{split} \tilde{\vec{\omega}}^M&=\left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}}\tilde{\vec{\omega}}^F\left[ Q_{\mathrm{M}}^{F} \right] \\ \tilde{\vec{\omega}}^F&=\left[ Q_{\mathrm{M}}^{F} \right] \tilde{\vec{\omega}}^M\left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}} \end{split} ω ~Mω ~F=[QMF]Tω ~F[QMF]=[QMF]ω ~M[QMF]T
进而可将上式中的项term [ Q ˙ M F ] R ⃗ P i M \left[ \dot{Q}{\mathrm{M}}^{F} \right] \vec{R}{\mathrm{P}{\mathrm{i}}}^{M} [Q˙MF]R PiM改写为（**下式仅当 P i P{\mathrm{i}} Pi为刚体上的固定点时成立** ）：
[ Q ˙ M F ] R ⃗ P i M = { ω ⃗ ~ F [ Q M F ] R ⃗ P i M = ω ⃗ ~ F R ⃗ P i F = ω ⃗ F × R ⃗ P i F [ Q M F ] ω ⃗ ~ M R ⃗ P i M = [ Q M F ] ( ω ⃗ M × R ⃗ P i M ) \left[ \dot{Q}{\mathrm{M}}^{F} \right] \vec{R}{\mathrm{P}{\mathrm{i}}}^{M}=\begin{cases} \tilde{\vec{\omega}}^F\left[ Q{\mathrm{M}}^{F} \right] \vec{R}{\mathrm{P}{\mathrm{i}}}^{M}=\tilde{\vec{\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] \tilde{\vec{\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{cases} [Q˙MF]R PiM=⎩ ⎨ ⎧ω ~F[QMF]R PiM=ω ~FR PiF=ω F×R PiF[QMF]ω ~MR PiM=[QMF](ω M×R PiM)
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 × 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−q4q1q4−q3−q4q1q2q3−q2q1 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−q4q1−q4q3q4q1−q2−q3q2q1 , 带入同样的式子可得：
ω ⃗ ~ F = 2 B ˉ B ˉ ˙ T ω ⃗ ~ M = 2 B B ˙ T \begin{split} \tilde{\vec{\omega}}^F&=2\bar{B}\dot{\bar{B}}^{\mathrm{T}} \\ \tilde{\vec{\omega}}^M&=2B\dot{B}^{\mathrm{T}} \end{split} ω ~Fω ~M=2BˉBˉ˙T=2BB˙T
将上式展开，由四元数的归一化可知： q ˙ 1 q 1 + q ˙ 2 q 2 + q ˙ 3 q 3 + q ˙ 4 q 4 = 0 \dot{q}_1q_1+\dot{q}_2q_2+\dot{q}_3q_3+\dot{q}_4q_4=0 q˙1q1+q˙2q2+q˙3q3+q˙4q4=0，可得：
[ 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 ] [ w 1 M w 2 M w 3 M ] = 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 ] \begin{split} \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] \\ \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{split} w1Fw2Fw3F w1Mw2Mw3M =2 q˙4q3−q˙3q4+q˙2q1−q˙1q2q˙2q4−q˙1q3+q˙4q2−q˙3q1q˙3q2−q˙4q1+q˙1q4−q˙2q3 =2 q4q˙3−q3q˙4−q2q˙1+q1q˙2q2q˙4+q1q˙3−q4q˙2−q3q˙1q3q˙2−q4q˙1+q1q˙4−q2q˙3
继续观察上式，将上式进行化简：
ω ⃗ F = 2 B q ⃗ ˙ = − 2 B ˙ q ⃗ ω ⃗ M = 2 B ˉ q ⃗ ˙ = − 2 B ˉ ˙ q ⃗ \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} ω F=2Bq ˙=−2B˙q ω M=2Bˉq ˙=−2Bˉ˙q
进而可将 [ Q ˙ M F ] R ⃗ P i M = [ Q M F ] ( ω ⃗ M × R ⃗ P i M ) \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) [Q˙MF]R PiM=[QMF](ω M×R PiM)改写为（下式仅当 P i P_{\mathrm{i}} Pi为刚体上的固定点时成立）：
[ Q ˙ M F ] R ⃗ P i M = [ Q M F ] ( ω ⃗ M × R ⃗ P i M ) = − [ Q M F ] ( R ⃗ P i M × ω ⃗ M ) = − [ Q M F ] R ⃗ ~ P i M ( 2 B ˉ q ⃗ ˙ ) \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] \tilde{\vec{R}}{\mathrm{P}{\mathrm{i}}}^{M}\left( 2\bar{B}\dot{\vec{q}} \right) [Q˙MF]R PiM=[QMF](ω M×R PiM)=−[QMF](R PiM×ω M)=−[QMF]R ~PiM(2Bˉq ˙)
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：
∂ ( [ Q M F ] R ⃗ P i M ) ∂ q ⃗ = − [ Q M F ] R ⃗ ~ P i M ( 2 B ˉ ) \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] \tilde{\vec{R}}{\mathrm{P}{\mathrm{i}}}^{M}\left( 2\bar{B} \right) ∂q ∂([QMF]R PiM)=−[QMF]R ~PiM(2Bˉ)
4.1.2 轴角参数表示角速度
将 [ θ v 1 v 2 v 3 ] = [ 2 a r c cos ( q 1 ) q 2 sin θ 2 q 3 sin θ 2 q 4 sin θ 2 ] \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] θv1v2v3 = 2arccos(q1)sin2θq2sin2θq3sin2θq4 带入 [ 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 ] , [ w 1 M w 2 M w 3 M ] = 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] , \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] w1Fw2Fw3F =2 q˙4q3−q˙3q4+q˙2q1−q˙1q2q˙2q4−q˙1q3+q˙4q2−q˙3q1q˙3q2−q˙4q1+q˙1q4−q˙2q3 , w1Mw2Mw3M =2 q4q˙3−q3q˙4−q2q˙1+q1q˙2q2q˙4+q1q˙3−q4q˙2−q3q˙1q3q˙2−q4q˙1+q1q˙4−q2q˙3 可得：
[ w 1 F w 2 F w 3 F ] = [ 2 ( v ˙ 3 v 2 − v ˙ 2 v 3 ) sin 2 θ 2 + v ˙ 1 sin θ + θ ˙ v 1 2 ( v ˙ 1 v 3 − v ˙ 3 v 1 ) sin 2 θ 2 + v ˙ 2 sin θ + θ ˙ v 2 2 ( v ˙ 2 v 1 − v ˙ 1 v 2 ) sin 2 θ 2 + v ˙ 3 sin θ + θ ˙ v 3 ] [ w 1 M w 2 M w 3 M ] = [ 2 ( v 3 v ˙ 2 − v 2 v ˙ 3 ) sin 2 θ 2 + v ˙ 1 sin θ + θ ˙ v 1 2 ( v 1 v ˙ 3 − v 3 v ˙ 1 ) sin 2 θ 2 + v ˙ 2 sin θ + θ ˙ v 2 2 ( v 2 v ˙ 1 − v 1 v ˙ 2 ) sin 2 θ 2 + v ˙ 3 sin θ + θ ˙ v 3 ] \begin{split} \left[ \begin{array}{c} {w_1}^F\\ {w_2}^F\\ {w_3}^F\\ \end{array} \right] &=\left[ \begin{array}{c} 2\left( \dot{v}_3v_2-\dot{v}_2v_3 \right) \sin ^2\frac{\theta}{2}+\dot{v}_1\sin \theta +\dot{\theta}v_1\\ 2\left( \dot{v}_1v_3-\dot{v}_3v_1 \right) \sin ^2\frac{\theta}{2}+\dot{v}_2\sin \theta +\dot{\theta}v_2\\ 2\left( \dot{v}_2v_1-\dot{v}_1v_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{split} w1Fw2Fw3F w1Mw2Mw3M = 2(v˙3v2−v˙2v3)sin22θ+v˙1sinθ+θ˙v12(v˙1v3−v˙3v1)sin22θ+v˙2sinθ+θ˙v22(v˙2v1−v˙1v2)sin22θ+v˙3sinθ+θ˙v3 = 2(v3v˙2−v2v˙3)sin22θ+v˙1sinθ+θ˙v12(v1v˙3−v3v˙1)sin22θ+v˙2sinθ+θ˙v22(v2v˙1−v1v˙2)sin22θ+v˙3sinθ+θ˙v3
整理为：
ω ⃗ F = 2 v ⃗ F × v ⃗ ˙ F sin 2 θ 2 + v ⃗ ˙ F sin θ + θ ˙ v ⃗ F ω ⃗ M = 2 v ⃗ ˙ F × v ⃗ F sin 2 θ 2 + v ⃗ ˙ F sin θ + θ ˙ v ⃗ F \begin{split} \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{split} ω Fω M=2v F×v ˙Fsin22θ+v ˙Fsinθ+θ˙v F=2v ˙F×v Fsin22θ+v ˙Fsinθ+θ˙v F
4.1.3 轴角参数表示角速度
对于ZYX欧拉角而言，有：
{ [ Q M F ] = [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] [ Q F 1 F ( i ⃗ F , α ) ] ω ⃗ ~ F = [ Q ˙ M F ] [ Q M F ] T ω ⃗ ~ F = { [ Q ˙ F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] [ Q F 1 F ( i ⃗ F , α ) ] ⋅ [ Q F 1 F ( i ⃗ F , α ) ] T [ Q F 2 F 1 ( j ⃗ F , β ) ] T [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] T + [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q ˙ F 2 F 1 ( j ⃗ F , β ) ] [ Q F 1 F ( i ⃗ F , α ) ] ⋅ [ Q F 1 F ( i ⃗ F , α ) ] T [ Q F 2 F 1 ( j ⃗ F , β ) ] T [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] T + [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] [ Q ˙ F 1 F ( i ⃗ F , α ) ] ⋅ [ Q F 1 F ( i ⃗ F , α ) ] T [ Q F 2 F 1 ( j ⃗ F , β ) ] T [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] T ω ⃗ ~ F = { [ Q ˙ F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] T + [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q ˙ F 2 F 1 ( j ⃗ F , β ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] T [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] T + [ [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] ] [ Q ˙ F 1 F ( i ⃗ F , α ) ] ⋅ [ Q F 1 F ( i ⃗ F , α ) ] T [ [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] ] T ω ⃗ ~ F = ω ⃗ ~ F 3 ( M ) F 2 + [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] ω ⃗ ~ F 2 F 1 ~ + [ [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] ] ω ⃗ ~ F 1 F ~ ⇒ ω ⃗ F = ω ⃗ F 3 ( M ) F 2 + [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] ω ⃗ F 2 F 1 + [ [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] ] ω ⃗ F 1 F ⇒ ω ⃗ F = [ 0 0 γ ˙ ] + [ cos γ − sin γ 0 sin γ cos γ 0 0 0 1 ] [ 0 β ˙ 0 ] + [ cos γ − sin γ 0 sin γ cos γ 0 0 0 1 ] [ cos β 0 sin β 0 1 0 − sin β 0 cos β ] [ α ˙ 0 0 ] ⇒ ω ⃗ F = [ cos β cos γ − sin γ 0 cos β sin γ cos γ 0 − sin β 0 1 ] [ α ˙ β ˙ γ ˙ ] \begin{split} &\begin{cases} \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]\\ \tilde{\vec{\omega}}^F=\left[ \dot{Q}{\mathrm{M}}^{F} \right] \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}}\\ \end{cases} \\ \tilde{\vec{\omega}}^F&=\begin{cases} \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{cases} \\ \tilde{\vec{\omega}}^F&=\begin{cases} \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{cases} \\ \tilde{\vec{\omega}}^F&=\tilde{\vec{\omega}}{\mathrm{F}3\left( M \right)}^{F_2}+\widetilde{\left[ Q{\mathrm{F}3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \tilde{\vec{\omega}}{\mathrm{F}2}^{F_1}}+\widetilde{\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] \tilde{\vec{\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{matrix} \cos \gamma& -\sin \gamma& 0\\ \sin \gamma& \cos \gamma& 0\\ 0& 0& 1\\ \end{matrix} \right] \left[ \begin{array}{c} 0\\ \dot{\beta}\\ 0\\ \end{array} \right] +\left[ \begin{matrix} \cos \gamma& -\sin \gamma& 0\\ \sin \gamma& \cos \gamma& 0\\ 0& 0& 1\\ \end{matrix} \right] \left[ \begin{matrix} \cos \beta& 0& \sin \beta\\ 0& 1& 0\\ -\sin \beta& 0& \cos \beta\\ \end{matrix} \right] \left[ \begin{array}{c} \dot{\alpha}\\ 0\\ 0\\ \end{array} \right] \\ \Rightarrow \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] \end{split} ω ~Fω ~Fω ~F⇒ω F⇒ω F⇒ω F⎩ ⎨ ⎧[QMF]=[QF3(M)F2(k F,γ)][QF2F1(j F,β)][QF1F(i F,α)]ω ~F=[Q˙MF][QMF]T=⎩ ⎨ ⎧[Q˙F3(M)F2(k F,γ)][QF2F1(j F,β)][QF1F(i F,α)]⋅[QF1F(i F,α)]T[QF2F1(j F,β)]T[QF3(M)F2(k F,γ)]T+[QF3(M)F2(k F,γ)][Q˙F2F1(j F,β)][QF1F(i F,α)]⋅[QF1F(i F,α)]T[QF2F1(j F,β)]T[QF3(M)F2(k F,γ)]T+[QF3(M)F2(k F,γ)][QF2F1(j F,β)][Q˙F1F(i F,α)]⋅[QF1F(i F,α)]T[QF2F1(j F,β)]T[QF3(M)F2(k F,γ)]T=⎩ ⎨ ⎧[Q˙F3(M)F2(k F,γ)][QF3(M)F2(k F,γ)]T+[QF3(M)F2(k F,γ)][Q˙F2F1(j F,β)][QF2F1(j F,β)]T[QF3(M)F2(k F,γ)]T+[[QF3(M)F2(k F,γ)][QF2F1(j F,β)]][Q˙F1F(i F,α)]⋅[QF1F(i F,α)]T[[QF3(M)F2(k F,γ)][QF2F1(j F,β)]]T=ω ~F3(M)F2+[QF3(M)F2(k F,γ)]ω ~F2F1 +[[QF3(M)F2(k F,γ)][QF2F1(j F,β)]]ω ~F1F =ω F3(M)F2+[QF3(M)F2(k F,γ)]ω F2F1+[[QF3(M)F2(k F,γ)][QF2F1(j F,β)]]ω F1F= 00γ˙ + cosγsinγ0−sinγcosγ0001 0β˙0 + cosγsinγ0−sinγcosγ0001 cosβ0−sinβ010sinβ0cosβ α˙00 = cosβcosγcosβsinγ−sinβ−sinγcosγ0001 α˙β˙γ˙