[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

---

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 ΣM​F​=mtotal​⋅R GF​×V GF​+∫(R GPi​F​⋅R GPi​F​)ω MF​dmi​−∫(R GPi​F​⋅ω MF​)R GPi​F​dmi​，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 ΣM​F​=mtotal​⋅R GF​×V GF​+∫(R GPi​F​TR GPi​F​⋅E3×3−R GPi​F​R GPi​F​T)dmi​⋅ω MF​=mtotal​⋅R GF​×V GF​+[I]ΣM​/GF​⋅ω MF​H ΣM​/GF​=H ΣM​F​−mtotal​⋅R GF​×V GF​=[I]ΣM​/GF​⋅ω MF​​
  则相对于质心点 G G G 存在：
  { M ⃗ Σ M / G F = [ I ] Σ M / G F α ⃗ M F + ω ⃗ M F × ( [ I ] Σ M / G F ⋅ ω ⃗ M F ) [ I ] Σ M / 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 F ⃗ G F = m t o t a l a ⃗ G F \begin{cases} \vec{M}_{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}=\left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\vec{\alpha}_{\mathrm{M}}^{F}+\vec{\omega}_{\mathrm{M}}^{F}\times \left( \left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F}\cdot \vec{\omega}_{\mathrm{M}}^{F} \right)\\ \left[ I \right] _{\Sigma _{\mathrm{M}}/\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}}\\ \vec{F}_{\mathrm{G}}^{F}=m_{\mathrm{total}}\vec{a}_{\mathrm{G}}^{F}\\ \end{cases} ⎩ ⎨ ⎧​M ΣM​/GF​=[I]ΣM​/GF​α MF​+ω MF​×([I]ΣM​/GF​⋅ω MF​)[I]ΣM​/GF​=∫(R GPi​F​TR GPi​F​⋅E3×3−R GPi​F​R GPi​F​T)dmi​F GF​=mtotal​a GF​​
• 对 H ⃗ Σ M / O F \vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F} H ΣM​/OF​进一步推导（分析 [ I ] Σ M / G F \left[ I \right] _{\Sigma _{\mathrm{M}}/\mathrm{G}}^{F} [I]ΣM​/GF​），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 × ( V ⃗ O 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 ⋅ R ⃗ ~ O P i F ⋅ V ⃗ O 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 ] ) + m t o t a l ⋅ R ⃗ ~ O G F ⋅ V ⃗ O 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 ] + m t o t a l ⋅ R ⃗ ~ O G F ⋅ V ⃗ O 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 ] + m t o t a l ⋅ R ⃗ ~ O G F ⋅ V ⃗ O 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 ] + m t o t a l ⋅ R ⃗ ~ O G F ⋅ V ⃗ O 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 ] + m t o t a l ⋅ R ⃗ ~ O G F ⋅ V ⃗ O F = [ I ^ J ^ K ^ ] T [ H x H y H z ] + m t o t a l ⋅ R ⃗ ~ O G F ⋅ V ⃗ O F \begin{aligned} \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{V}_{\mathrm{O}}^{F}+\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 \tilde{\vec{R}}_{\mathrm{OP}_{\mathrm{i}}}^{F}\cdot \vec{V}_{\mathrm{O}}^{F}}\\ &=\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) +m_{\mathrm{total}}\cdot \tilde{\vec{R}}_{\mathrm{OG}}^{F}\cdot \vec{V}_{\mathrm{O}}^{F}}\\ &=\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] +m_{\mathrm{total}}\cdot \tilde{\vec{R}}_{\mathrm{OG}}^{F}\cdot \vec{V}_{\mathrm{O}}^{F}}\\ &=\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] +m_{\mathrm{total}}\cdot \tilde{\vec{R}}_{\mathrm{OG}}^{F}\cdot \vec{V}_{\mathrm{O}}^{F}}\\ &=\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] +m_{\mathrm{total}}\cdot \tilde{\vec{R}}_{\mathrm{OG}}^{F}\cdot \vec{V}_{\mathrm{O}}^{F}\,\,\\ &=\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] +m_{\mathrm{total}}\cdot \tilde{\vec{R}}_{\mathrm{OG}}^{F}\cdot \vec{V}_{\mathrm{O}}^{F}\\ &=\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] +m_{\mathrm{total}}\cdot \tilde{\vec{R}}_{\mathrm{OG}}^{F}\cdot \vec{V}_{\mathrm{O}}^{F}\\ \end{aligned} H ΣM​/OF​​=i∑N​R OPi​F​×P Pi​F​=i∑N​mPi​​⋅R OPi​F​×(V OF​+ω F×R OPi​F​)=i∑N​mPi​​⋅R ~OPi​F​⋅(ω ~F⋅R OPi​F​)+i∑N​mPi​​⋅R ~OPi​F​⋅V OF​=i∑N​mPi​​⋅ ​I^J^K^​ ​T ​0zOPi​F​−yOPi​F​​−zOPi​F​0xOPi​F​​yOPi​F​−xOPi​F​0​ ​⋅ ​I^J^K^​ ​T ​ ​0wzPi​F​−wyPi​F​​−wzPi​F​0wxPi​F​​wyPi​F​−wxPi​F​0​ ​⋅ ​xOPi​F​yOPi​F​zOPi​F​​ ​ ​+mtotal​⋅R ~OGF​⋅V OF​=i∑N​mPi​​⋅ ​I^J^K^​ ​T ​[(yOPi​F​)2+(zOPi​F​)2]wxPi​F​−(xOPi​F​yOPi​F​)wyPi​F​−(xOPi​F​zOPi​F​)wzPi​F​−(yOPi​F​xOPi​F​)wxPi​F​+[(xOPi​F​)2+(zOPi​F​)2]wyPi​F​−(yOPi​F​zOPi​F​)wzPi​F​−(zOPi​F​xOPi​F​)wxPi​F​−(zOPi​F​yOPi​F​)wyPi​F​+[(xOPi​F​)2+(yOPi​F​)2]wzPi​F​​ ​+mtotal​⋅R ~OGF​⋅V OF​=i∑N​mPi​​⋅ ​I^J^K^​ ​T ​(yOPi​F​)2+(zOPi​F​)2−yOPi​F​xOPi​F​−zOPi​F​xOPi​F​​−xOPi​F​yOPi​F​(xOPi​F​)2+(zOPi​F​)2−zOPi​F​yOPi​F​​−xOPi​F​zOPi​F​−yOPi​F​zOPi​F​(xOPi​F​)2+(yOPi​F​)2​ ​ ​wxPi​F​wyPi​F​wzPi​F​​ ​+mtotal​⋅R ~OGF​⋅V OF​= ​I^J^K^​ ​T ​∑iN​mPi​​⋅[(yOPi​F​)2+(zOPi​F​)2]−∑iN​mPi​​⋅(yOPi​F​xOPi​F​)−∑iN​mPi​​⋅(zOPi​F​xOPi​F​)​−∑iN​mPi​​⋅xOPi​F​yOPi​F​∑iN​mPi​​⋅[(xOPi​F​)2+(zOPi​F​)2]−∑iN​mPi​​⋅(zOPi​F​yOPi​F​)​−∑iN​mPi​​⋅(xOPi​F​zOPi​F​)−∑iN​mPi​​⋅(yOPi​F​zOPi​F​)∑iN​mPi​​⋅[(xOPi​F​)2+(yOPi​F​)2]​ ​ ​wxPi​F​wyPi​F​wzPi​F​​ ​+mtotal​⋅R ~OGF​⋅V OF​= ​I^J^K^​ ​T ​Ixx​Iyx​Izx​​Ixy​Iyy​Izy​​Ixz​Iyz​Izz​​ ​ ​wxPi​F​wyPi​F​wzPi​F​​ ​= ​I^J^K^​ ​T ​Ixx​wxPi​F​+Ixy​wyPi​F​+Ixz​wzPi​F​Iyx​wxPi​F​+Iyy​wyPi​F​+Iyz​wzPi​F​Izx​wxPi​F​+Izy​wyPi​F​+Izz​wzPi​F​​ ​+mtotal​⋅R ~OGF​⋅V OF​= ​I^J^K^​ ​T ​Hx​Hy​Hz​​ ​+mtotal​⋅R ~OGF​⋅V OF​​

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：（为方便运算，忽略点 O O O的运动）

• 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 1​⋅r 3​)r 2​−(r 2​⋅r 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 ] \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]} H ΣM​/OF​=i∑N​R OPi​F​×P Pi​F​=i∑N​mPi​​⋅R OPi​F​×(ω F×R OPi​F​)=i∑N​mPi​​⋅[(R OPi​F​⋅R OPi​F​)ω F−(ω F⋅R OPi​F​)R OPi​F​]