[足式机器人]Part3 机构运动学与动力学分析与建模 Ch00-2(3) 质量刚体的在坐标系下运动

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:
黎 旭,陈 强 洪,甄 文 强 等.惯 性 张 量 平 移 和 旋 转 复 合 变 换 的 一 般 形 式 及 其 应 用[J].工 程 数 学 学 报,2022,39(06):1005-1011.

食用方法
质量点的动量与角动量
刚体的动量与角动量——力与力矩的关系
惯性矩阵的表达与推导——在刚体运动过程中的作用
惯性矩阵在不同坐标系下的表达
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 Ch00-2质量刚体的在坐标系下运动Part3

      • 2.2.3 欧拉方程 Euler equation - 2


2.2.3 欧拉方程 Euler equation - 2

  • 进而分析 H ⃗ Σ M F = m t o t a l ⋅ R ⃗ G F × V ⃗ G F + ∫ ( R ⃗ G P i F ⋅ R ⃗ G P i F ) ω ⃗ M F d m i − ∫ ( R ⃗ G P i F ⋅ ω ⃗ M F ) R ⃗ G P i F d m i \vec{H}_{\Sigma _{\mathrm{M}}}^{F}=m_{\mathrm{total}}\cdot \vec{R}_{\mathrm{G}}^{F}\times \vec{V}_{\mathrm{G}}^{F}+\int{\left( \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F} \right) \vec{\omega}_{\mathrm{M}}^{F}}\mathrm{d}m_{\mathrm{i}}-\int{\left( \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right) \vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}\mathrm{d}m_{\mathrm{i}} H ΣMF=mtotalR GF×V GF+(R GPiFR GPiF)ω MFdmi(R GPiFω MF)R GPiFdmi,有:
    H ⃗ Σ M F = m t o t a l ⋅ R ⃗ G F × V ⃗ G F + ∫ ( R ⃗ G P i F T R ⃗ G P i F ⋅ E 3 × 3 − R ⃗ G P i F R ⃗ G P i F T ) d m i ⋅ ω ⃗ M F = m t o t a l ⋅ R ⃗ G F × V ⃗ G F + [ I ] Σ M / G F ⋅ ω ⃗ M F H ⃗ Σ M / G F = H ⃗ Σ M F − m t o t a l ⋅ R ⃗ G F × V ⃗ G F = [ I ] Σ M / G F ⋅ ω ⃗ M F \begin{split} &\vec{H}_{\Sigma _{\mathrm{M}}}^{F}=m_{\mathrm{total}}\cdot \vec{R}_{\mathrm{G}}^{F}\times \vec{V}_{\mathrm{G}}^{F}+\int{\left( {\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}}\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}\cdot E^{3\times 3}-\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}{\vec{R}_{\mathrm{GP}_{\mathrm{i}}}^{F}}^{\mathrm{T}} \right)}\mathrm{d}m_{\mathrm{i}}\cdot \vec{\omega}_{\mathrm{M}}^{F} =m_{\mathrm{total}}\cdot \vec{R}_{\mathrm{G}}^{F}\times \vec{V}_{\mathrm{G}}^{F}+\left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \\ &\vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}=\vec{H}_{\Sigma _{\mathrm{M}}}^{F}-m_{\mathrm{total}}\cdot \vec{R}_{\mathrm{G}}^{F}\times \vec{V}_{\mathrm{G}}^{F}=\left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \end{split} H ΣMF=mtotalR GF×V GF+(R GPiFTR GPiFE3×3R GPiFR GPiFT)dmiω MF=mtotalR GF×V GF+[I]ΣM/GFω MFH ΣM/GF=H ΣMFmtotalR GF×V GF=[I]ΣM/GFω MF
    则相对于质心点 G G G 存在:
    { τ ⃗ G F = d h ⃗ G F d t τ ⃗ G / O F = d h ⃗ G / O F d t + V ⃗ O F × P ⃗ G F P ⃗ G F = m t o t a l V ⃗ G F \begin{cases} \vec{\tau}_{\mathrm{G}}^{F}=\frac{\mathrm{d}\vec{h}_{\mathrm{G}}^{F}}{\mathrm{dt}}\\ \vec{\tau}_{\mathrm{G}/\mathrm{O}}^{F}=\frac{\mathrm{d}\vec{h}_{\mathrm{G}/\mathrm{O}}^{F}}{\mathrm{dt}}+\vec{V}_{\mathrm{O}}^{F}\times \vec{P}_{\mathrm{G}}^{F}\\ \vec{P}_{\mathrm{G}}^{F}=m_{\mathrm{total}}\vec{V}_{\mathrm{G}}^{F}\\ \end{cases} τ GF=dtdh GFτ G/OF=dtdh G/OF+V OF×P GFP GF=mtotalV GF
  • H ⃗ Σ M / O F \vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F} H ΣM/OF进一步推导,可得:
    H ⃗ Σ M / O F = ∑ i N R ⃗ O P i F × P ⃗ P i F = ∑ i N m P i ⋅ R ⃗ O P i F × ( ω ⃗ F × R ⃗ O P i F ) = ∑ i N m P i ⋅ R ⃗ ~ O P i F ⋅ ( ω ⃗ ~ F ⋅ R ⃗ O P i F ) = ∑ i N m P i ⋅ [ I ^ J ^ K ^ ] T [ 0 − z O P i F y O P i F z O P i F 0 − x O P i F − y O P i F x O P i F 0 ] ⋅ [ I ^ J ^ K ^ ] T ( [ 0 − w z P i F w y P i F w z P i F 0 − w x P i F − w y P i F w x P i F 0 ] ⋅ [ x O P i F y O P i F z O P i F ] ) = ∑ i N m P i ⋅ [ I ^ J ^ K ^ ] T [ [ ( y O P i F ) 2 + ( z O P i F ) 2 ] w x P i F − ( x O P i F y O P i F ) w y P i F − ( x O P i F z O P i F ) w z P i F − ( y O P i F x O P i F ) w x P i F + [ ( x O P i F ) 2 + ( z O P i F ) 2 ] w y P i F − ( y O P i F z O P i F ) w z P i F − ( z O P i F x O P i F ) w x P i F − ( z O P i F y O P i F ) w y P i F + [ ( x O P i F ) 2 + ( y O P i F ) 2 ] w z P i F ] = ∑ i N m P i ⋅ [ I ^ J ^ K ^ ] T [ ( y O P i F ) 2 + ( z O P i F ) 2 − x O P i F y O P i F − x O P i F z O P i F − y O P i F x O P i F ( x O P i F ) 2 + ( z O P i F ) 2 − y O P i F z O P i F − z O P i F x O P i F − z O P i F y O P i F ( x O P i F ) 2 + ( y O P i F ) 2 ] [ w x P i F w y P i F w z P i F ] = [ I ^ J ^ K ^ ] T [ ∑ i N m P i ⋅ [ ( y O P i F ) 2 + ( z O P i F ) 2 ] − ∑ i N m P i ⋅ x O P i F y O P i F − ∑ i N m P i ⋅ ( x O P i F z O P i F ) − ∑ i N m P i ⋅ ( y O P i F x O P i F ) ∑ i N m P i ⋅ [ ( x O P i F ) 2 + ( z O P i F ) 2 ] − ∑ i N m P i ⋅ ( y O P i F z O P i F ) − ∑ i N m P i ⋅ ( z O P i F x O P i F ) − ∑ i N m P i ⋅ ( z O P i F y O P i F ) ∑ i N m P i ⋅ [ ( x O P i F ) 2 + ( y O P i F ) 2 ] ] [ w x P i F w y P i F w z P i F ]    = [ I ^ J ^ K ^ ] T [ I x x I x y I x z I y x I y y I y z I z x I z y I z z ] [ w x P i F w y P i F w z P i F ] = [ I ^ J ^ K ^ ] T [ I x x w x P i F + I x y w y P i F + I x z w z P i F I y x w x P i F + I y y w y P i F + I y z w z P i F I z x w x P i F + I z y w y P i F + I z z w z P i F ] = [ I ^ J ^ K ^ ] T [ H x H y H z ] \begin{split} \vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F}&=\sum_i^N{\vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F}\times \vec{P}_{\mathrm{P}_{\mathrm{i}}}^{F}}=\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F}\times \left( \vec{\omega}^F\times \vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}=\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \tilde{\vec{R}}_{\mathrm{OP}_{\mathrm{i}}}^{F}\cdot \left( \tilde{\vec{\omega}}^F\cdot \vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)} \\ &=\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{matrix} 0& -z_{\mathrm{OP}_{\mathrm{i}}}^{F}& y_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ z_{\mathrm{OP}_{\mathrm{i}}}^{F}& 0& -x_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ -y_{\mathrm{OP}_{\mathrm{i}}}^{F}& x_{\mathrm{OP}_{\mathrm{i}}}^{F}& 0\\ \end{matrix} \right] \cdot \left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left( \left[ \begin{matrix} 0& -w_{\mathrm{z}_{\mathrm{Pi}}}^{F}& w_{\mathrm{y}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{z}_{\mathrm{Pi}}}^{F}& 0& -w_{\mathrm{x}_{\mathrm{Pi}}}^{F}\\ -w_{\mathrm{y}_{\mathrm{Pi}}}^{F}& w_{\mathrm{x}_{\mathrm{Pi}}}^{F}& 0\\ \end{matrix} \right] \cdot \left[ \begin{array}{c} x_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ y_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ z_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ \end{array} \right] \right)} \\ &=\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} \left[ \left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right] w_{\mathrm{x}_{\mathrm{Pi}}}^{F}-\left( x_{\mathrm{OP}_{\mathrm{i}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) w_{\mathrm{y}_{\mathrm{Pi}}}^{F}-\left( x_{\mathrm{OP}_{\mathrm{i}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ -\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) w_{\mathrm{x}_{\mathrm{Pi}}}^{F}+\left[ \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right] w_{\mathrm{y}_{\mathrm{Pi}}}^{F}-\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ -\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) w_{\mathrm{x}_{\mathrm{Pi}}}^{F}-\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) w_{\mathrm{y}_{\mathrm{Pi}}}^{F}+\left[ \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right] w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right]} \\ &=\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{matrix} \left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2& -x_{\mathrm{OP}_{\mathrm{i}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F}& -x_{\mathrm{OP}_{\mathrm{i}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ -y_{\mathrm{OP}_{\mathrm{i}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F}& \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2& -y_{\mathrm{OP}_{\mathrm{i}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ -z_{\mathrm{OP}_{\mathrm{i}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F}& -z_{\mathrm{OP}_{\mathrm{i}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F}& \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2\\ \end{matrix} \right] \left[ \begin{array}{c} w_{\mathrm{x}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{y}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right]} \\ &=\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{matrix} \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right]}& -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot x_{\mathrm{OP}_{\mathrm{i}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F}}& -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}\\ -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( y_{\mathrm{OP}_{\mathrm{i}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}& \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right]}& -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( y_{\mathrm{OP}_{\mathrm{i}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}\\ -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( z_{\mathrm{OP}_{\mathrm{i}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}& -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( z_{\mathrm{OP}_{\mathrm{i}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}& \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right]}\\ \end{matrix} \right] \left[ \begin{array}{c} w_{\mathrm{x}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{y}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right] \,\, \\ &=\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{matrix} I_{\mathrm{xx}}& I_{\mathrm{xy}}& I_{\mathrm{xz}}\\ I_{\mathrm{yx}}& I_{\mathrm{yy}}& I_{\mathrm{yz}}\\ I_{\mathrm{zx}}& I_{\mathrm{zy}}& I_{\mathrm{zz}}\\ \end{matrix} \right] \left[ \begin{array}{c} w_{\mathrm{x}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{y}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right] =\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} I_{\mathrm{xx}}w_{\mathrm{x}_{\mathrm{Pi}}}^{F}+I_{\mathrm{xy}}w_{\mathrm{y}_{\mathrm{Pi}}}^{F}+I_{\mathrm{xz}}w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ I_{\mathrm{yx}}w_{\mathrm{x}_{\mathrm{Pi}}}^{F}+I_{\mathrm{yy}}w_{\mathrm{y}_{\mathrm{Pi}}}^{F}+I_{\mathrm{yz}}w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ I_{\mathrm{zx}}w_{\mathrm{x}_{\mathrm{Pi}}}^{F}+I_{\mathrm{zy}}w_{\mathrm{y}_{\mathrm{Pi}}}^{F}+I_{\mathrm{zz}}w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right] \\ &=\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} H_{\mathrm{x}}\\ H_{\mathrm{y}}\\ H_{\mathrm{z}}\\ \end{array} \right] \end{split} H ΣM/OF=iNR OPiF×P PiF=iNmPiR OPiF×(ω F×R OPiF)=iNmPiR ~OPiF(ω ~FR OPiF)=iNmPi I^J^K^ T 0zOPiFyOPiFzOPiF0xOPiFyOPiFxOPiF0 I^J^K^ T 0wzPiFwyPiFwzPiF0wxPiFwyPiFwxPiF0 xOPiFyOPiFzOPiF =iNmPi I^J^K^ T [(yOPiF)2+(zOPiF)2]wxPiF(xOPiFyOPiF)wyPiF(xOPiFzOPiF)wzPiF(yOPiFxOPiF)wxPiF+[(xOPiF)2+(zOPiF)2]wyPiF(yOPiFzOPiF)wzPiF(zOPiFxOPiF)wxPiF(zOPiFyOPiF)wyPiF+[(xOPiF)2+(yOPiF)2]wzPiF =iNmPi I^J^K^ T (yOPiF)2+(zOPiF)2yOPiFxOPiFzOPiFxOPiFxOPiFyOPiF(xOPiF)2+(zOPiF)2zOPiFyOPiFxOPiFzOPiFyOPiFzOPiF(xOPiF)2+(yOPiF)2 wxPiFwyPiFwzPiF = I^J^K^ T iNmPi[(yOPiF)2+(zOPiF)2]iNmPi(yOPiFxOPiF)iNmPi(zOPiFxOPiF)iNmPixOPiFyOPiFiNmPi[(xOPiF)2+(zOPiF)2]iNmPi(zOPiFyOPiF)iNmPi(xOPiFzOPiF)iNmPi(yOPiFzOPiF)iNmPi[(xOPiF)2+(yOPiF)2] wxPiFwyPiFwzPiF = I^J^K^ T IxxIyxIzxIxyIyyIzyIxzIyzIzz wxPiFwyPiFwzPiF = I^J^K^ T IxxwxPiF+IxywyPiF+IxzwzPiFIyxwxPiF+IyywyPiF+IyzwzPiFIzxwxPiF+IzywyPiF+IzzwzPiF = I^J^K^ T HxHyHz

其中:

  • 若有: ω ⃗ = [ I ^ J ^ K ^ ] T [ ω 1 ω 2 ω 3 ] , R ⃗ = [ I ^ J ^ K ^ ] T [ r 1 r 2 r 3 ] \vec{\omega}=\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} \omega _1\\ \omega _2\\ \omega _3\\ \end{array} \right] ,\vec{R}=\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} r_1\\ r_2\\ r_3\\ \end{array} \right] ω = I^J^K^ T ω1ω2ω3 ,R = I^J^K^ T r1r2r3 ,则有如下叉乘的计算:
    ω ⃗ × R ⃗ = ω ⃗ ~ ⋅ R ⃗ = [ I ^ J ^ K ^ ] T ( [ 0 − ω 3 ω 2 ω 3 0 − ω 1 − ω 2 ω 1 0 ] ⋅ [ r 1 r 2 r 3 ] ) \vec{\omega}\times \vec{R}=\tilde{\vec{\omega}}\cdot \vec{R}=\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left( \left[ \begin{matrix} 0& -\omega _3& \omega _2\\ \omega _3& 0& -\omega _1\\ -\omega _2& \omega _1& 0\\ \end{matrix} \right] \cdot \left[ \begin{array}{c} r_1\\ r_2\\ r_3\\ \end{array} \right] \right) ω ×R =ω ~R = I^J^K^ T 0ω3ω2ω30ω1ω2ω10 r1r2r3
  • H ⃗ Σ M / O F \vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F} H ΣM/OF表示刚体 Σ M \Sigma _{\mathrm{M}} ΣM相对于(with respect to/W.R.T) O O O 的角动量在固定坐标系 { F } \left\{ F \right\} {F}的表达。其投影分量满足:
    [ H x H y H z ] = [ I x x w x P i F + I x y w y P i F + I x z w z P i F I y x w x P i F + I y y w y P i F + I y z w z P i F I z x w x P i F + I z y w y P i F + I z z w z P i F ] = [ I x x I x y I x z I y x I y y I y z I z x I z y I z z ] [ w x P i F w y P i F w z P i F ] = [ I ] [ w x P i F w y P i F w z P i F ] \left[ \begin{array}{c} H_{\mathrm{x}}\\ H_{\mathrm{y}}\\ H_{\mathrm{z}}\\ \end{array} \right] =\left[ \begin{array}{c} I_{\mathrm{xx}}w_{\mathrm{x}_{\mathrm{Pi}}}^{F}+I_{\mathrm{xy}}w_{\mathrm{y}_{\mathrm{Pi}}}^{F}+I_{\mathrm{xz}}w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ I_{\mathrm{yx}}w_{\mathrm{x}_{\mathrm{Pi}}}^{F}+I_{\mathrm{yy}}w_{\mathrm{y}_{\mathrm{Pi}}}^{F}+I_{\mathrm{yz}}w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ I_{\mathrm{zx}}w_{\mathrm{x}_{\mathrm{Pi}}}^{F}+I_{\mathrm{zy}}w_{\mathrm{y}_{\mathrm{Pi}}}^{F}+I_{\mathrm{zz}}w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right] =\left[ \begin{matrix} I_{\mathrm{xx}}& I_{\mathrm{xy}}& I_{\mathrm{xz}}\\ I_{\mathrm{yx}}& I_{\mathrm{yy}}& I_{\mathrm{yz}}\\ I_{\mathrm{zx}}& I_{\mathrm{zy}}& I_{\mathrm{zz}}\\ \end{matrix} \right] \left[ \begin{array}{c} w_{\mathrm{x}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{y}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right] =\left[ I \right] \left[ \begin{array}{c} w_{\mathrm{x}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{y}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right] HxHyHz = IxxwxPiF+IxywyPiF+IxzwzPiFIyxwxPiF+IyywyPiF+IyzwzPiFIzxwxPiF+IzywyPiF+IzzwzPiF = IxxIyxIzxIxyIyyIzyIxzIyzIzz wxPiFwyPiFwzPiF =[I] wxPiFwyPiFwzPiF
  • 矩阵 [ I ] \left[ I \right] [I]常被称为{惯性矩阵Inertia-matrix,有: H ⃗ Σ M / O F = [ I ] ω ⃗ F \vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F}=\left[ I \right] \vec{\omega}^F H ΣM/OF=[I]ω F,其中:
    [ I ] = [ I x x I x y I x z I y x I y y I y z I z x I z y I z z ] = [ ∑ i N m P i ⋅ [ ( y O P i F ) 2 + ( z O P i F ) 2 ] − ∑ i N m P i ⋅ x O P i F y O P i F − ∑ i N m P i ⋅ ( x O P i F z O P i F ) − ∑ i N m P i ⋅ ( y O P i F x O P i F ) ∑ i N m P i ⋅ [ ( x O P i F ) 2 + ( z O P i F ) 2 ] − ∑ i N m P i ⋅ ( y O P i F z O P i F ) − ∑ i N m P i ⋅ ( z O P i F x O P i F ) − ∑ i N m P i ⋅ ( z O P i F y O P i F ) ∑ i N m P i ⋅ [ ( x O P i F ) 2 + ( y O P i F ) 2 ] ] \begin{split} \left[ I \right] &=\left[ \begin{matrix} I_{\mathrm{xx}}& I_{\mathrm{xy}}& I_{\mathrm{xz}}\\ I_{\mathrm{yx}}& I_{\mathrm{yy}}& I_{\mathrm{yz}}\\ I_{\mathrm{zx}}& I_{\mathrm{zy}}& I_{\mathrm{zz}}\\ \end{matrix} \right] \\ &=\left[ \begin{matrix} \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right]}& -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot x_{\mathrm{OP}_{\mathrm{i}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F}}& -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}\\ -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( y_{\mathrm{OP}_{\mathrm{i}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}& \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right]}& -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( y_{\mathrm{OP}_{\mathrm{i}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}\\ -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( z_{\mathrm{OP}_{\mathrm{i}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}& -\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left( z_{\mathrm{OP}_{\mathrm{i}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}& \sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right]}\\ \end{matrix} \right] \end{split} [I]= IxxIyxIzxIxyIyyIzyIxzIyzIzz = iNmPi[(yOPiF)2+(zOPiF)2]iNmPi(yOPiFxOPiF)iNmPi(zOPiFxOPiF)iNmPixOPiFyOPiFiNmPi[(xOPiF)2+(zOPiF)2]iNmPi(zOPiFyOPiF)iNmPi(xOPiFzOPiF)iNmPi(yOPiFzOPiF)iNmPi[(xOPiF)2+(yOPiF)2]

上式的实际推导过程,是进行两次转置变化,在实际过程中可以理解成,适用于矩阵与矢量相乘的张量Tensor乘法,因此也可将惯性矩阵 [ I ] \left[ I \right] [I]称为惯性张量Inertia Tensor。而采用基于拉格朗日恒等式证明的三个向量的双重矢积公式,可能更利于理解:

  • 三个向量的双重矢积公式: ( r ⃗ 1 × r ⃗ 2 ) × r ⃗ 3 = ( r ⃗ 1 ⋅ r ⃗ 3 ) r ⃗ 2 − ( r ⃗ 2 ⋅ r ⃗ 3 ) r ⃗ 1 \left( \vec{r}_1\times \vec{r}_2 \right) \times \vec{r}_3=\left( \vec{r}_1\cdot \vec{r}_3 \right) \vec{r}_2-\left( \vec{r}_2\cdot \vec{r}_3 \right) \vec{r}_1 (r 1×r 2)×r 3=(r 1r 3)r 2(r 2r 3)r 1
    H ⃗ Σ M / O F = ∑ i N R ⃗ O P i F × P ⃗ P i F = ∑ i N m P i ⋅ R ⃗ O P i F × ( ω ⃗ F × R ⃗ O P i F ) = ∑ i N m P i ⋅ [ ( R ⃗ O P i F ⋅ R ⃗ O P i F ) ω ⃗ F − ( ω ⃗ F ⋅ R ⃗ O P i F ) R ⃗ O P i F ] = ∑ i N m P i ⋅ [ I ^ J ^ K ^ ] T [ ( [ x O P i F y O P i F z O P i F ] T [ x O P i F y O P i F z O P i F ] ) [ w x P i F w y P i F w z P i F ] − ( [ w x P i F w y P i F w z P i F ] T [ x O P i F y O P i F z O P i F ] ) [ x O P i F y O P i F z O P i F ] ] = ∑ i N m P i ⋅ [ I ^ J ^ K ^ ] T [ [ ( ( x O P i F ) 2 + ( y O P i F ) 2 + ( z O P i F ) 2 ) w x P i F ( ( x O P i F ) 2 + ( y O P i F ) 2 + ( z O P i F ) 2 ) w y P i F ( ( x O P i F ) 2 + ( y O P i F ) 2 + ( z O P i F ) 2 ) w z P i F ] − [ ( w x P i F x O P i F + w y P i F y O P i F + w z P i F z O P i F ) x O P i F ( w x P i F x O P i F + w y P i F y O P i F + w z P i F z O P i F ) y O P i F ( w x P i F x O P i F + w y P i F y O P i F + w z P i F z O P i F ) z O P i F ] ] = ∑ i N m P i ⋅ [ I ^ J ^ K ^ ] T [ [ ( y O P i F ) 2 + ( z O P i F ) 2 ] w x P i F − ( x O P i F y O P i F ) w y P i F − ( x O P i F z O P i F ) w z P i F − ( y O P i F x O P i F ) w x P i F + [ ( x O P i F ) 2 + ( z O P i F ) 2 ] w y P i F − ( y O P i F z O P i F ) w z P i F − ( z O P i F x O P i F ) w x P i F − ( z O P i F y O P i F ) w y P i F + [ ( x O P i F ) 2 + ( y O P i F ) 2 ] w z P i F ] = ∑ i N m P i ⋅ [ I ^ J ^ K ^ ] T [ ( y O P i F ) 2 + ( z O P i F ) 2 − x O P i F y O P i F − x O P i F z O P i F − y O P i F x O P i F ( x O P i F ) 2 + ( z O P i F ) 2 − y O P i F z O P i F − z O P i F x O P i F − z O P i F y O P i F ( x O P i F ) 2 + ( y O P i F ) 2 ] [ w x P i F w y P i F w z P i F ] \begin{split} \vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F}&=\sum_i^N{\vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F}\times \vec{P}_{\mathrm{P}_{\mathrm{i}}}^{F}}=\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F}\times \left( \vec{\omega}^F\times \vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F} \right)}=\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \left( \vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F}\cdot \vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) \vec{\omega}^F-\left( \vec{\omega}^F\cdot \vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) \vec{R}_{\mathrm{OP}_{\mathrm{i}}}^{F} \right]} \\ &=\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \left( \left[ \begin{array}{c} x_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ y_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ z_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} x_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ y_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ z_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ \end{array} \right] \right) \left[ \begin{array}{c} w_{\mathrm{x}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{y}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right] -\left( \left[ \begin{array}{c} w_{\mathrm{x}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{y}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} x_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ y_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ z_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ \end{array} \right] \right) \left[ \begin{array}{c} x_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ y_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ z_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ \end{array} \right] \right]} \\ &=\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \left[ \begin{array}{c} \left( \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right) w_{\mathrm{x}_{\mathrm{Pi}}}^{F}\\ \left( \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right) w_{\mathrm{y}_{\mathrm{Pi}}}^{F}\\ \left( \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right) w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right] -\left[ \begin{array}{c} \left( w_{\mathrm{x}_{\mathrm{Pi}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F}+w_{\mathrm{y}_{\mathrm{Pi}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F}+w_{\mathrm{z}_{\mathrm{Pi}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) x_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ \left( w_{\mathrm{x}_{\mathrm{Pi}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F}+w_{\mathrm{y}_{\mathrm{Pi}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F}+w_{\mathrm{z}_{\mathrm{Pi}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) y_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ \left( w_{\mathrm{x}_{\mathrm{Pi}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F}+w_{\mathrm{y}_{\mathrm{Pi}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F}+w_{\mathrm{z}_{\mathrm{Pi}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) z_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ \end{array} \right] \right]} \\ &=\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}\left[ \begin{array}{c} \left[ \left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right] w_{\mathrm{x}_{\mathrm{Pi}}}^{F}-\left( x_{\mathrm{OP}_{\mathrm{i}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) w_{\mathrm{y}_{\mathrm{Pi}}}^{F}-\left( x_{\mathrm{OP}_{\mathrm{i}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ -\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) w_{\mathrm{x}_{\mathrm{Pi}}}^{F}+\left[ \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right] w_{\mathrm{y}_{\mathrm{Pi}}}^{F}-\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ -\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) w_{\mathrm{x}_{\mathrm{Pi}}}^{F}-\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) w_{\mathrm{y}_{\mathrm{Pi}}}^{F}+\left[ \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2 \right] w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right]} \\ &=\sum_i^N{m_{\mathrm{P}_{\mathrm{i}}}\cdot \left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] ^{\mathrm{T}}}\left[ \begin{matrix} \left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2& -x_{\mathrm{OP}_{\mathrm{i}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F}& -x_{\mathrm{OP}_{\mathrm{i}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ -y_{\mathrm{OP}_{\mathrm{i}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F}& \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( z_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2& -y_{\mathrm{OP}_{\mathrm{i}}}^{F}z_{\mathrm{OP}_{\mathrm{i}}}^{F}\\ -z_{\mathrm{OP}_{\mathrm{i}}}^{F}x_{\mathrm{OP}_{\mathrm{i}}}^{F}& -z_{\mathrm{OP}_{\mathrm{i}}}^{F}y_{\mathrm{OP}_{\mathrm{i}}}^{F}& \left( x_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2+\left( y_{\mathrm{OP}_{\mathrm{i}}}^{F} \right) ^2\\ \end{matrix} \right] \left[ \begin{array}{c} w_{\mathrm{x}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{y}_{\mathrm{Pi}}}^{F}\\ w_{\mathrm{z}_{\mathrm{Pi}}}^{F}\\ \end{array} \right] \end{split} H ΣM/OF=iNR OPiF×P PiF=iNmPiR OPiF×(ω F×R OPiF)=iNmPi[(R OPiFR OPiF)ω F(ω FR OPiF)R OPiF]=iNmPi I^J^K^ T xOPiFyOPiFzOPiF T xOPiFyOPiFzOPiF wxPiFwyPiFwzPiF wxPiFwyPiFwzPiF

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