[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)

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=mtotal⋅R GF×V GF+∫(R GPiF⋅R GPiF)ω MFdmi−∫(R GPiF⋅ω 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=mtotal⋅R GF×V GF+∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmi⋅ω MF=mtotal⋅R GF×V GF+ $I$ ΣM/GF⋅ω MFH ΣM/GF=H ΣMF−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/GFα MF+ω MF×( $I$ ΣM/GF⋅ω MF) $I$ ΣM/GF=∫(R GPiFTR GPiF⋅E3×3−R GPiFR GPiFT)dmiF GF=mtotala 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∑NR OPiF×P PiF=i∑NmPi⋅R OPiF×(V OF+ω F×R OPiF)=i∑NmPi⋅R ~OPiF⋅(ω ~F⋅R OPiF)+i∑NmPi⋅R ~OPiF⋅V OF=i∑NmPi⋅ I^J^K^ T 0zOPiF−yOPiF−zOPiF0xOPiFyOPiF−xOPiF0 ⋅ I^J^K^ T 0wzPiF−wyPiF−wzPiF0wxPiFwyPiF−wxPiF0 ⋅ xOPiFyOPiFzOPiF +mtotal⋅R ~OGF⋅V OF=i∑NmPi⋅ 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 +mtotal⋅R ~OGF⋅V OF=i∑NmPi⋅ I^J^K^ T (yOPiF)2+(zOPiF)2−yOPiFxOPiF−zOPiFxOPiF−xOPiFyOPiF(xOPiF)2+(zOPiF)2−zOPiFyOPiF−xOPiFzOPiF−yOPiFzOPiF(xOPiF)2+(yOPiF)2 wxPiFwyPiFwzPiF +mtotal⋅R ~OGF⋅V OF= I^J^K^ T ∑iNmPi⋅ $(yOPiF)2+(zOPiF)2$ −∑iNmPi⋅(yOPiFxOPiF)−∑iNmPi⋅(zOPiFxOPiF)−∑iNmPi⋅xOPiFyOPiF∑iNmPi⋅ $(xOPiF)2+(zOPiF)2$ −∑iNmPi⋅(zOPiFyOPiF)−∑iNmPi⋅(xOPiFzOPiF)−∑iNmPi⋅(yOPiFzOPiF)∑iNmPi⋅ $(xOPiF)2+(yOPiF)2$ wxPiFwyPiFwzPiF +mtotal⋅R ~OGF⋅V OF= I^J^K^ T IxxIyxIzxIxyIyyIzyIxzIyzIzz wxPiFwyPiFwzPiF = I^J^K^ T IxxwxPiF+IxywyPiF+IxzwzPiFIyxwxPiF+IyywyPiF+IyzwzPiFIzxwxPiF+IzywyPiF+IzzwzPiF +mtotal⋅R ~OGF⋅V OF= I^J^K^ T HxHyHz +mtotal⋅R ~OGF⋅V OF

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 0ω3−ω2−ω30ω1ω2−ω10 ⋅ 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)点 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 = $I$ Σ M / O F ω ⃗ F \vec{H}_{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F}=\left $I \\right$ _{\Sigma _{\mathrm{M}}/\mathrm{O}}^{F}\vec{\omega}^F H ΣM/OF= $I$ ΣM/OFω F，其中： $I \] Σ M / O F = \[ I x x Σ M / O F I x y Σ M / O F I x z Σ M / O F I y x Σ M / O F I y y Σ M / O F I y z Σ M / O F I z x Σ M / O F I z y Σ M / O F I z z Σ M / O F \] = \[ ∑ 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{aligned} \\left\[ I \\right\] _{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\&=\\left\[ \\begin{matrix} {I_{\\mathrm{xx}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\& {I_{\\mathrm{xy}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\& {I_{\\mathrm{xz}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\\\\ {I_{\\mathrm{yx}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\& {I_{\\mathrm{yy}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\& {I_{\\mathrm{yz}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\\\\ {I_{\\mathrm{zx}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\& {I_{\\mathrm{zy}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\& {I_{\\mathrm{zz}}}_{\\Sigma _{\\mathrm{M}}/\\mathrm{O}}\^{F}\\\\ \\end{matrix} \\right\]\\\\ \&=\\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{aligned} \[I\]ΣM/OF= IxxΣM/OFIyxΣM/OFIzxΣM/OFIxyΣM/OFIyyΣM/OFIzyΣM/OFIxzΣM/OFIyzΣM/OFIzzΣM/OF = ∑iNmPi⋅\[(yOPiF)2+(zOPiF)2\]−∑iNmPi⋅(yOPiFxOPiF)−∑iNmPi⋅(zOPiFxOPiF)−∑iNmPi⋅xOPiFyOPiF∑iNmPi⋅\[(xOPiF)2+(zOPiF)2\]−∑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：（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 ⋅ $( R ⃗ O P i F \cdot R ⃗ O P i F ) ω ⃗ F - ( ω ⃗ F \cdot R ⃗ O P i F ) R ⃗ O P i F$ \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∑NR OPiF×P PiF=i∑NmPi⋅R OPiF×(ω F×R OPiF)=i∑NmPi⋅ $(R OPiF\cdotR OPiF)ω F-(ω F\cdotR OPiF)R OPiF$