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 Part2
5. 运动刚体的加速度与角加速度
对 v ⃗ P i M = ( ω ⃗ ~ M − [ Q M F ] T [ Q ˙ M F ] ) R ⃗ P i M \vec{v}_{\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} v PiM=(ω ~M−[QMF]T[Q˙MF])R PiM 进一步求导，可计算出其运动刚体上点 P i P_i Pi的加速度为：
v ⃗ P i F = v ⃗ M F + [ Q ˙ M F ] R ⃗ P i M + [ Q M F ] R ⃗ ˙ P i M = v ⃗ M F + ω ⃗ ~ F [ Q M F ] R ⃗ P i M + [ Q M F ] v ⃗ P i M ⇒ a ⃗ P i F = a ⃗ M F + ( ω ⃗ ~ ˙ F [ Q M F ] R ⃗ P i M + ω ⃗ ~ F [ Q ˙ M F ] R ⃗ P i M + ω ⃗ ~ F [ Q M F ] v ⃗ P i M ) + ( [ Q ˙ M F ] v ⃗ P i M + [ Q M F ] a ⃗ P i M ) ⇒ a ⃗ P i F = a ⃗ M F + ( α ⃗ ~ F [ Q M F ] R ⃗ P i M + ω ⃗ ~ F ω ⃗ ~ F [ Q M F ] R ⃗ P i M + ω ⃗ ~ F [ Q M F ] v ⃗ P i M ) + ( ω ⃗ ~ F [ Q M F ] v ⃗ P i M + [ Q M F ] a ⃗ P i M ) ⇒ a ⃗ P i F = a ⃗ M F + α ⃗ ~ F ( R ⃗ P i M ) F + ω ⃗ ~ F ω ⃗ ~ F ( R ⃗ P i M ) F + 2 ω ⃗ ~ F ( v ⃗ P i M ) F + ( a ⃗ P i M ) F \begin{split} &\vec{v}_{\mathrm{P}_{\mathrm{i}}}^{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{v}_{\mathrm{M}}^{F}+\tilde{\vec{\omega}}^F\left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M}+\left[ Q_{\mathrm{M}}^{F} \right] \vec{v}_{\mathrm{P}_{\mathrm{i}}}^{M} \\ \Rightarrow \vec{a}_{\mathrm{P}_{\mathrm{i}}}^{F}&=\vec{a}_{\mathrm{M}}^{F}+\left( \dot{\tilde{\vec{\omega}}}^F\left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M}+\tilde{\vec{\omega}}^F\left[ \dot{Q}_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M}+\tilde{\vec{\omega}}^F\left[ Q_{\mathrm{M}}^{F} \right] \vec{v}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) +\left( \left[ \dot{Q}_{\mathrm{M}}^{F} \right] \vec{v}_{\mathrm{P}_{\mathrm{i}}}^{M}+\left[ Q_{\mathrm{M}}^{F} \right] \vec{a}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) \\ \Rightarrow \vec{a}_{\mathrm{P}_{\mathrm{i}}}^{F}&=\vec{a}_{\mathrm{M}}^{F}+\left( \tilde{\vec{\alpha}}^F\left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M}+\tilde{\vec{\omega}}^F\tilde{\vec{\omega}}^F\left[ Q_{\mathrm{M}}^{F} \right] \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M}+\tilde{\vec{\omega}}^F\left[ Q_{\mathrm{M}}^{F} \right] \vec{v}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) +\left( \tilde{\vec{\omega}}^F\left[ Q_{\mathrm{M}}^{F} \right] \vec{v}_{\mathrm{P}_{\mathrm{i}}}^{M}+\left[ Q_{\mathrm{M}}^{F} \right] \vec{a}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) \\ \Rightarrow \vec{a}_{\mathrm{P}_{\mathrm{i}}}^{F}&=\vec{a}_{\mathrm{M}}^{F}+\tilde{\vec{\alpha}}^F\left( \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^F+\tilde{\vec{\omega}}^F\tilde{\vec{\omega}}^F\left( \vec{R}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^F+2\tilde{\vec{\omega}}^F\left( \vec{v}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^F+\left( \vec{a}_{\mathrm{P}_{\mathrm{i}}}^{M} \right) ^F \end{split} ⇒a PiF⇒a PiF⇒a PiFv PiF=v MF+[Q˙MF]R