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:
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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
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
- 4.2 速度传递
- 4.3 运动刚体的加速度与角加速度
- 4.3.1 欧拉角表示角加速度
- 4.3.2 待补充
- 4.3 刚体运动的旋量表达方式
4.2 速度传递
对于连续转动的坐标系而言,有:
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\begin{split} &\left\{ \begin{array}{c} \left[ Q_{\mathrm{M}}^{F} \right] =\left[ Q_{\mathrm{F}_1}^{F} \right] \left[ Q_{\mathrm{F}_2}^{F_1} \right] \cdots \left[ Q_{\mathrm{M}}^{F_{\mathrm{n}-1}} \right]\\ \tilde{\vec{\omega}}^F=\left[ \dot{Q}_{\mathrm{M}}^{F} \right] \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}}\\ \end{array} \right. \\ \Rightarrow \tilde{\vec{\omega}}^F&=\left( \left[ \dot{Q}_{\mathrm{F}_1}^{F} \right] \left[ Q_{\mathrm{F}_2}^{F_1} \right] \cdots \left[ Q_{\mathrm{M}}^{F_{\mathrm{n}-1}} \right] +\left[ Q_{\mathrm{F}_1}^{F} \right] \left[ \dot{Q}_{\mathrm{F}_2}^{F_1} \right] \cdots \left[ Q_{\mathrm{M}}^{F_{\mathrm{n}-1}} \right] +\cdots +\left[ Q_{\mathrm{F}_1}^{F} \right] \left[ Q_{\mathrm{F}_2}^{F_1} \right] \cdots \left[ \dot{Q}_{\mathrm{M}}^{F_{\mathrm{n}-1}} \right] \right) \cdot \left[ Q_{\mathrm{M}}^{F_{\mathrm{n}-1}} \right] ^{\mathrm{T}}\cdots \left[ Q_{\mathrm{F}_2}^{F_1} \right] ^{\mathrm{T}}\left[ Q_{\mathrm{F}_1}^{F} \right] ^{\mathrm{T}} \\ \Rightarrow \tilde{\vec{\omega}}^F&=\left[ \dot{Q}_{\mathrm{F}_1}^{F} \right] \left[ Q_{\mathrm{F}_1}^{F} \right] ^{\mathrm{T}}+\left[ Q_{\mathrm{F}_1}^{F} \right] \left[ \dot{Q}_{\mathrm{F}_2}^{F_1} \right] \left[ Q_{\mathrm{F}_2}^{F_1} \right] ^{\mathrm{T}}\left[ Q_{\mathrm{F}_1}^{F} \right] ^{\mathrm{T}}+\left[ Q_{\mathrm{F}_2}^{F} \right] \left[ \dot{Q}_{\mathrm{F}_3}^{F_2} \right] \left[ Q_{\mathrm{F}_3}^{F_2} \right] ^{\mathrm{T}}\left[ Q_{\mathrm{F}_2}^{F} \right] ^{\mathrm{T}}+\cdots \left[ Q_{\mathrm{F}_{\mathrm{n}-1}}^{F} \right] \left[ \dot{Q}_{\mathrm{M}}^{F_{\mathrm{n}-1}} \right] \left[ Q_{\mathrm{M}}^{F_{\mathrm{n}-1}} \right] ^{\mathrm{T}}\left[ Q_{\mathrm{F}_{\mathrm{n}-1}}^{F} \right] ^{\mathrm{T}} \\ \Rightarrow \tilde{\vec{\omega}}^F&=\tilde{\vec{\omega}}_{\mathrm{F}_1}^{F}+\widetilde{\left[ Q_{\mathrm{F}_1}^{F} \right] \vec{\omega}_{\mathrm{F}_2}^{F_1}}+\widetilde{\left[ Q_{\mathrm{F}_2}^{F} \right] \vec{\omega}_{\mathrm{F}_3}^{F_2}}+\cdots +\widetilde{\left[ Q_{\mathrm{F}_{\mathrm{n}-1}}^{F} \right] \vec{\omega}_{\mathrm{M}}^{F_{\mathrm{n}-1}}} \\ \Rightarrow \vec{\omega}^F&=\vec{\omega}_{\mathrm{F}_1}^{F}+\left[ Q_{\mathrm{F}_1}^{F} \right] \vec{\omega}_{\mathrm{F}_2}^{F_1}+\cdots +\left[ Q_{\mathrm{F}_{\mathrm{n}-1}}^{F} \right] \vec{\omega}_{\mathrm{M}}^{F_{\mathrm{n}-1}} \end{split}
⇒ω~F⇒ω~F⇒ω~F⇒ωF⎩
⎨
⎧[QMF]=[QF1F][QF2F1]⋯[QMFn−1]ω~F=[Q˙MF][QMF]T=([Q˙F1F][QF2F1]⋯[QMFn−1]+[QF1F][Q˙F2F1]⋯[QMFn−1]+⋯+[QF1F][QF2F1]⋯[Q˙MFn−1])⋅[QMFn−1]T⋯[QF2F1]T[QF1F]T=[Q˙F1F][QF1F]T+[QF1F][Q˙F2F1][QF2F1]T[QF1F]T+[QF2F][Q˙F3F2][QF3F2]T[QF2F]T+⋯[QFn−1F][Q˙MFn−1][QMFn−1]T[QFn−1F]T=ω~F1F+[QF1F]ωF2F1
+[QF2F]ωF3F2
+⋯+[QFn−1F]ωMFn−1
=ωF1F+[QF1F]ωF2F1+⋯+[QFn−1F]ωMFn−1
此时,
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\vec{\omega}_{\mathrm{F}_1}^{F}
ωF1F理解为,坐标系
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{F1}所代表的刚体在坐标系
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{F}下的角速度参数。
4.3 运动刚体的加速度与角加速度
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\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}
vPiM=(ω~M−[QMF]T[Q˙MF])RPiM 进一步求导,可计算出其运动刚体上点
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\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}
⇒aPiF⇒aPiF⇒aPiFvPiF=vMF+[Q˙MF]RPiM+[QMF]R˙PiM=vMF+ω~F[QMF]RPiM+[QMF]vPiM=aMF+(ω~˙F[QMF]RPiM+ω~F[Q˙MF]RPiM+ω~F[QMF]vPiM)+([Q˙MF]vPiM+[QMF]aPiM)=aMF+(α~F[QMF]RPiM+ω~Fω~F[QMF]RPiM+ω~F[QMF]vPiM)+(ω~F[QMF]vPiM+[QMF]aPiM)=aMF+α~F(RPiM)F+ω~Fω~F(RPiM)F+2ω~F(vPiM)F+(aPiM)F
当点
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\begin{split} \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}}_{\nearrow 0} \right) ^F+\left( {\vec{a}_{\mathrm{P}_{\mathrm{i}}}^{M}}_{\nearrow 0} \right) ^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 \end{split}
aPiF=aMF+α~F(RPiM)F+ω~Fω~F(RPiM)F+2ω~F(vPiM↗0)F+(aPiM↗0)F=aMF+α~F(RPiM)F+ω~Fω~F(RPiM)F
4.3.1 欧拉角表示角加速度
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\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]
ωF=
cosβcosγcosβsinγ−sinβ−sinγcosγ0001
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\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}
ωF=
cosβcosγcosβsinγ−sinβ−sinγcosγ0001
α˙β˙γ˙
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−sinβcosγ−cosβsinγcosβcosγ−sinβsinγ−cosβ−cosγ−sinγ0000
α˙β˙γ˙
+
cosβcosγcosβsinγ−sinβ−sinγcosγ0001
α¨β¨γ¨