互联网站备案信息查询,保定seo外包服务商,wordpress使用手机号登录,查询商品价格走势的网站本文仅供学习使用#xff0c;总结很多本现有讲述运动学或动力学书籍后的总结#xff0c;从矢量的角度进行分析#xff0c;方法比较传统#xff0c;但更易理解#xff0c;并且现有的看似抽象方法#xff0c;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… 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4.1 角速度的表达4.1.1 欧拉参数表示角速度4.1.2 轴角参数表示角速度4.1.3 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 PFR MF[QMF]R PiM⇒v PFv MF[Q˙MF]R PiM[QMF]R ˙PiMω F×R PFω ~FR PFω ~F(R MF[QMF]R PiM)⇒[Q˙MF]R PiM[QMF]R ˙PiMω ~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 PiM0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]]T0 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 PiMω ~FR PiFω 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}}^F2\bar{B}\dot{\bar{B}}^{\mathrm{T}} \\ \tilde{\vec{\omega}}^M2B\dot{B}^{\mathrm{T}} \end{split} ω ~Fω ~M2BˉBˉ˙T2BB˙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_40 q˙1q1q˙2q2q˙3q3q˙4q40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}_1q_1\dot{q}_2\\ q_2\dot{q}_4q_1\dot{q}_3-q_4\dot{q}_2-q_3\dot{q}_1\\ q_3\dot{q}_2-q_4\dot{q}_1q_1\dot{q}_4-q_2\dot{q}_3\\ \end{array} \right] \end{split} w1Fw2Fw3F w1Mw2Mw3M 2 q˙4q3−q˙3q4q˙2q1−q˙1q2q˙2q4−q˙1q3q˙4q2−q˙3q1q˙3q2−q˙4q1q˙1q4−q˙2q3 2 q4q˙3−q3q˙4−q2q˙1q1q˙2q2q˙4q1q˙3−q4q˙2−q3q˙1q3q˙2−q4q˙1q1q˙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}^F2B\dot{\vec{q}}-2\dot{B}\vec{q} \\ \vec{\omega}^M2\bar{B}\dot{\vec{q}}-2\dot{\bar{B}}\vec{q} ω F2Bq ˙−2B˙q ω M2Bˉq ˙−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}_1q_1\dot{q}_2\\ q_2\dot{q}_4q_1\dot{q}_3-q_4\dot{q}_2-q_3\dot{q}_1\\ q_3\dot{q}_2-q_4\dot{q}_1q_1\dot{q}_4-q_2\dot{q}_3\\ \end{array} \right] w1Fw2Fw3F 2 q˙4q3−q˙3q4q˙2q1−q˙1q2q˙2q4−q˙1q3q˙4q2−q˙3q1q˙3q2−q˙4q1q˙1q4−q˙2q3 , w1Mw2Mw3M 2 q4q˙3−q3q˙4−q2q˙1q1q˙2q2q˙4q1q˙3−q4q˙2−q3q˙1q3q˙2−q4q˙1q1q˙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)sin22θv˙1sinθθ˙v12(v˙1v3−v˙3v1)sin22θv˙2sinθθ˙v22(v˙2v1−v˙1v2)sin22θv˙3sinθθ˙v3 2(v3v˙2−v2v˙3)sin22θv˙1sinθθ˙v12(v1v˙3−v3v˙1)sin22θv˙2sinθθ˙v22(v2v˙1−v1v˙2)sin22θv˙3sinθθ˙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}^F2\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}^M2\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 M2v F×v ˙Fsin22θv ˙Fsinθθ˙v F2v ˙F×v Fsin22θv ˙Fsinθθ˙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)]]Tω ~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˙
