[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

---

2.2.3 欧拉方程 Euler equation - 2

• 进而分析 $H_{{\Sigma }_{\mathrm{M}}}^F=m_{\mathrm{total}}\cdot R_{\mathrm{G}}^F\times V_{\mathrm{G}}^F+\int \left(R_{{\mathrm{GP}}_{\mathrm{i}}}^F\cdot R_{{\mathrm{GP}}_{\mathrm{i}}}^F\right){\omega }_{\mathrm{M}}^F\mathrm{d}{m}_{\mathrm{i}}-\int \left(R_{{\mathrm{GP}}_{\mathrm{i}}}^F\cdot {\omega }_{\mathrm{M}}^F\right)R_{{\mathrm{GP}}_{\mathrm{i}}}^F\mathrm{d}{m}_{\mathrm{i}}$，有：
  $$\begin{array}{rl}& H_{{\Sigma }_{\mathrm{M}}}^F=m_{\mathrm{total}}\cdot R_{\mathrm{G}}^F\times V_{\mathrm{G}}^F+\int \left({R_{{\mathrm{GP}}_{\mathrm{i}}}^F}^{\mathrm{T}}{R}_{{\mathrm{GP}}_{\mathrm{i}}}^F\cdot E^{3\times 3}-R_{{\mathrm{GP}}_{\mathrm{i}}}^F{R_{{\mathrm{GP}}_{\mathrm{i}}}^F}^{\mathrm{T}}\right)\mathrm{d}{m}_{\mathrm{i}}\cdot {\omega }_{\mathrm{M}}^F=m_{\mathrm{total}}\cdot R_{\mathrm{G}}^F\times V_{\mathrm{G}}^F+{\left[I\right]}_{{\Sigma }_{\mathrm{M}}/\mathrm{G}}^F\cdot {\omega }_{\mathrm{M}}^F\\ & H_{{\Sigma }_{\mathrm{M}}/\mathrm{G}}^F=H_{{\Sigma }_{\mathrm{M}}}^F-m_{\mathrm{total}}\cdot R_{\mathrm{G}}^F\times V_{\mathrm{G}}^F={\left[I\right]}_{{\Sigma }_{\mathrm{M}}/\mathrm{G}}^F\cdot {\omega }_{\mathrm{M}}^F\end{array}$$
  则相对于质心点 $G$ 存在：
  $$\{\begin{array}{l}{\tau }_{\mathrm{G}}^F=\frac{\mathrm{d}{h}_{\mathrm{G}}^F}{\mathrm{dt}}\\ {\tau }_{\mathrm{G}/\mathrm{O}}^F=\frac{\mathrm{d}{h}_{\mathrm{G}/\mathrm{O}}^F}{\mathrm{dt}}+V_{\mathrm{O}}^F\times P_{\mathrm{G}}^F\\ P_{\mathrm{G}}^F=m_{\mathrm{total}}{V}_{\mathrm{G}}^F\end{array}$$
• 对$H_{{\Sigma }_{\mathrm{M}}/\mathrm{O}}^F$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​​=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​⋅(ω ~F⋅R OPi​F​)=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​​ ​ ​=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​​ ​=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​​ ​= ​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​​ ​= ​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​​ ​= ​I^J^K^​ ​T ​Hx​Hy​Hz​​ ​​

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 ​ω1​ω2​ω3​​ ​,R = ​I^J^K^​ ​T ​r1​r2​r3​​ ​，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​​−ω3​0ω1​​ω2​−ω1​0​ ​⋅ ​r1​r2​r3​​ ​ ​
• $H_{{\Sigma }_{\mathrm{M}}/\mathrm{O}}^F$表示刚体${\Sigma }_{\mathrm{M}}$相对于(with respect to/W.R.T)点 $O$ 的角动量在固定坐标系$\left\{F\right\}$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] ​Hx​Hy​Hz​​ ​= ​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​​ ​= ​Ixx​Iyx​Izx​​Ixy​Iyy​Izy​​Ixz​Iyz​Izz​​ ​ ​wxPi​F​wyPi​F​wzPi​F​​ ​=[I] ​wxPi​F​wyPi​F​wzPi​F​​ ​
• 矩阵$\left[I\right]$常被称为{惯性矩阵Inertia-matrix，有：$H_{{\Sigma }_{\mathrm{M}}/\mathrm{O}}^F=\left[I\right]{\omega }^F$，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]​= ​Ixx​Iyx​Izx​​Ixy​Iyy​Izy​​Ixz​Iyz​Izz​​ ​= ​∑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]​ ​​

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$\left[I\right]$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：$\left(r_1\times r_2\right)\times r_3=\left(r_1\cdot r_3\right)r_2-\left(r_2\cdot r_3\right)r_1$
  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​​=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​]=i∑N​mPi​​⋅ ​I^J^K^​ ​T ​ ​ ​xOPi​F​yOPi​F​zOPi​F​​ ​T ​xOPi​F​yOPi​F​zOPi​F​​ ​ ​ ​wxPi​F​wyPi​F​wzPi​F​​ ​− ​ ​wxPi​F​wyPi​F​wzPi​F​​ ​