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
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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Part4
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- [2.2.4 牛顿-欧拉方程 Netwon-Euler equation](#2.2.4 牛顿-欧拉方程 Netwon-Euler equation)
- [2.3 惯性矩阵的转换 Inertia-Matrix Transformation](#2.3 惯性矩阵的转换 Inertia-Matrix Transformation)
- [2.4 惯性矩阵的主轴定理} Principal Axis Theorem](#2.4 惯性矩阵的主轴定理} Principal Axis Theorem)
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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{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{\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{aligned} H ΣM/OF=i∑NR OPiF×P PiF=i∑NmPi⋅R OPiF×(ω F×R OPiF)=i∑NmPi⋅[(R OPiF⋅R OPiF)ω F−(ω F⋅R OPiF)R OPiF]=i∑NmPi⋅ I^J^K^ T xOPiFyOPiFzOPiF T xOPiFyOPiFzOPiF wxPiFwyPiFwzPiF − wxPiFwyPiFwzPiF T xOPiFyOPiFzOPiF xOPiFyOPiFzOPiF =i∑NmPi⋅ I^J^K^ T ((xOPiF)2+(yOPiF)2+(zOPiF)2)wxPiF((xOPiF)2+(yOPiF)2+(zOPiF)2)wyPiF((xOPiF)2+(yOPiF)2+(zOPiF)2)wzPiF − (wxPiFxOPiF+wyPiFyOPiF+wzPiFzOPiF)xOPiF(wxPiFxOPiF+wyPiFyOPiF+wzPiFzOPiF)yOPiF(wxPiFxOPiF+wyPiFyOPiF+wzPiFzOPiF)zOPiF =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 =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
对 H ⃗ Σ M / O F \vec{H}{\Sigma {\mathrm{M}}/\mathrm{O}}^{F} H ΣM/OF进一步处理可得： H ⃗ Σ M / O 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 ) = ∑ i N m P i ⋅ R ⃗ ~ O P i F ( − R ⃗ ~ O P i F ) ω ⃗ F \vec{H}{\Sigma {\mathrm{M}}/\mathrm{O}}^{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 \vec{R}{\mathrm{OP}{\mathrm{i}}}^{F}\times \left( -\vec{R}{\mathrm{OP}{\mathrm{i}}}^{F}\times \vec{\omega}^F \right)}=\sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot \tilde{\vec{R}}{\mathrm{OP}{\mathrm{i}}}^{F}\left( -\tilde{\vec{R}}{\mathrm{OP}{\mathrm{i}}}^{F} \right)}\vec{\omega}^F H ΣM/OF=∑iNmPi⋅R OPiF×(ω F×R OPiF)=∑iNmPi⋅R OPiF×(−R OPiF×ω F)=∑iNmPi⋅R ~OPiF(−R ~OPiF)ω F。进而得出： ⇒ [ I ] = ∑ i N m P i ⋅ R ⃗ ~ O P i F ( − R ⃗ ~ O P i F ) \Rightarrow \left[ I \right] =\sum_i^N{m{\mathrm{P}{\mathrm{i}}}\cdot \tilde{\vec{R}}{\mathrm{OP}{\mathrm{i}}}^{F}\left( -\tilde{\vec{R}}{\mathrm{OP}_{\mathrm{i}}}^{F} \right)} ⇒[I]=∑iNmPi⋅R ~OPiF(−R ~OPiF)
2.2.4 牛顿-欧拉方程 Netwon-Euler equation
刚体动力学中常用：
{ F ⃗ Σ M F = m t o t a l ⋅ a ⃗ G F M ⃗ Σ M / G F = [ I ] Σ M / G F α ⃗ M F + ω ⃗ M F × ( [ I ] Σ M / G F ⋅ ω ⃗ M F ) \begin{cases} \vec{F}{\Sigma {\mathrm{M}}}^{F}=m{\mathrm{total}}\cdot \vec{a}{\mathrm{G}}^{F}\\ \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)\\ \end{cases} {F ΣMF=mtotal⋅a GFM ΣM/GF=[I]ΣM/GFα MF+ω MF×([I]ΣM/GF⋅ω MF)
2.3 惯性矩阵的转换 Inertia-Matrix Transformation
对于空间中的运动刚体而言，刚体的惯性矩阵一般会根据运动坐标系 { M } \left\{ M \right\} \,\, {M}的基矢量为基底进行计算，而不会直接考虑运动刚体在固定坐标系 { F } \left\{ F \right\} \,\, {F}下的惯性矩阵。此时运动坐标系 { M } \left\{ M \right\} \,\, {M}下计算得出的惯性矩阵记为： [ I ] M \left[ I \right] ^M [I]M。若运动坐标系 { M } \left\{ M \right\} \,\, {M}与固定坐标系 { F } \left\{ F \right\} \,\, {F}的基矢量满足： [ i ⃗ M j ⃗ M k ⃗ M ] = [ Q M F ] T [ I ^ J ^ K ^ ] \left[ \begin{array}{c} \vec{i}^M\\ \vec{j}^M\\ \vec{k}^M\\ \end{array} \right] =\left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}}\left[ \begin{array}{c} \hat{I}\\ \hat{J}\\ \hat{K}\\ \end{array} \right] i Mj Mk M =[QMF]T I^J^K^ ，其中 [ Q M F ] T \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}} [QMF]T为转换矩阵Transition Matrix，为正交矩阵Orthogonal Matrix（满足 [ Q M F ] T = [ Q M F ] − 1 = [ Q F M ] \left[ Q_{\mathrm{M}}^{F} \right] ^T=\left[ Q_{\mathrm{M}}^{F} \right] ^{-1}=\left[ Q_{\mathrm{F}}^{M} \right] [QMF]T=[QMF]−1=[QFM]）， [ Q M F ] \left[ Q_{\mathrm{M}}^{F} \right] [QMF]又称旋转矩阵Rotation~Matrix
（一个向量乘以一个正交阵，相当于对这个向量进行旋转）。也揭示了该矩阵的两个作用：基底转换 （转换矩阵 [ Q M F ] T \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}} [QMF]T）与向量旋转 （旋转矩阵 [ Q M F ] \left[ Q_{\mathrm{M}}^{F} \right] [QMF]），则考虑最开始的图有：
R ⃗ P i F = R ⃗ M F + [ Q M F ] R ⃗ P i M \vec{R}{\mathrm{P}{\mathrm{i}}}^{F}=\vec{R}{\mathrm{M}}^{F}+\left[ Q{\mathrm{M}}^{F} \right] \vec{R}{\mathrm{P}{\mathrm{i}}}^{M} R PiF=R MF+[QMF]R PiM
进而分析惯性矩阵，若 O O O 点与固定坐标系原点 F F F 重合，则有：
I \] Σ M F = ∑ i N m P i ⋅ \[ ( R ⃗ P i F ) T R ⃗ P i F ⋅ E − R ⃗ P i F ( R ⃗ P i F ) T \] = ∑ i N m P i ⋅ \[ ( R ⃗ M F + \[ Q M F \] R ⃗ P i M ) T ( R ⃗ M F + \[ Q M F \] R ⃗ P i M ) ⋅ E − ( R ⃗ M F + \[ Q M F \] R ⃗ P i M ) ( R ⃗ M F + \[ Q M F \] R ⃗ P i M ) T \] = { m t o t a l ⋅ \[ ( R ⃗ M F ) T R ⃗ M F ⋅ E − R ⃗ M F ( R ⃗ M F ) T \] ⏟ \[ I 1 \] Σ M F + \[ Q M F \] ( ∑ i N m P i ⋅ \[ ( R ⃗ P i M ) T R ⃗ P i M ⋅ E − R ⃗ P i M ( R ⃗ P i M ) T \] ) \[ Q M F \] T + ⏟ \[ I 2 \] Σ M F m t o t a l ⋅ \[ ( R ⃗ M F ) T ( \[ Q M F \] R ⃗ C o M M ) ⋅ E − R ⃗ M F ( \[ Q M F \] R ⃗ C o M M ) T \] ⏟ \[ I 3 \] Σ M F + m t o t a l ⋅ \[ ( \[ Q M F \] R ⃗ C o M M ) T R ⃗ M F ⋅ E − ( \[ Q M F \] R ⃗ C o M M ) ( R ⃗ M F ) T \] ⏟ \[ I 4 \] Σ M F = \[ I 1 \] Σ M F + \[ I 2 \] Σ M F + \[ I 3 \] Σ M F + \[ I 4 \] Σ M F \\begin{split} \\left\[ I \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F}\&=\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left\[ \\left( \\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{F} \\right) \^T\\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{F}\\cdot E-\\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{F}\\left( \\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{F} \\right) \^T \\right\]} \\\\ \&=\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left\[ \\left( \\vec{R}_{\\mathrm{M}}\^{F}+\\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^{\\mathrm{T}}\\left( \\vec{R}_{\\mathrm{M}}\^{F}+\\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \\cdot E-\\left( \\vec{R}_{\\mathrm{M}}\^{F}+\\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \\left( \\vec{R}_{\\mathrm{M}}\^{F}+\\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^{\\mathrm{T}} \\right\]} \\\\ \&=\\left\\{ \\begin{array}{c} \\begin{array}{c} \\underbrace{m_{\\mathrm{total}}\\cdot \\left\[ \\left( \\vec{R}_{\\mathrm{M}}\^{F} \\right) \^{\\mathrm{T}}\\vec{R}_{\\mathrm{M}}\^{F}\\cdot E-\\vec{R}_{\\mathrm{M}}\^{F}\\left( \\vec{R}_{\\mathrm{M}}\^{F} \\right) \^{\\mathrm{T}} \\right\] }\\\\ \\left\[ I_1 \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F}\\\\ \\end{array}+\\\\ \\begin{array}{c} \\underbrace{\\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\left( \\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left\[ \\left( \\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^{\\mathrm{T}}\\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M}\\cdot E-\\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M}\\left( \\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^{\\mathrm{T}} \\right\]} \\right) \\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \^{\\mathrm{T}}+}\\\\ \\left\[ I_2 \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F}\\\\ \\end{array}\\\\ \\begin{array}{c} \\underbrace{m_{\\mathrm{total}}\\cdot \\left\[ \\left( \\vec{R}_{\\mathrm{M}}\^{F} \\right) \^{\\mathrm{T}}\\left( \\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\vec{R}_{\\mathrm{CoM}}\^{M} \\right) \\cdot E-\\vec{R}_{\\mathrm{M}}\^{F}\\left( \\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\vec{R}_{\\mathrm{CoM}}\^{M} \\right) \^{\\mathrm{T}} \\right\] }\\\\ \\left\[ I_3 \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F}\\\\ \\end{array}+\\\\ \\begin{array}{c} \\underbrace{m_{\\mathrm{total}}\\cdot \\left\[ \\left( \\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\vec{R}_{\\mathrm{CoM}}\^{M} \\right) \^T\\vec{R}_{\\mathrm{M}}\^{F}\\cdot E-\\left( \\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\vec{R}_{\\mathrm{CoM}}\^{M} \\right) \\left( \\vec{R}_{\\mathrm{M}}\^{F} \\right) \^{\\mathrm{T}} \\right\] }\\\\ \\left\[ I_4 \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F}\\\\ \\end{array}\\\\ \\end{array} \\right. \\\\ \&=\\left\[ I_1 \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F}+\\left\[ I_2 \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F}+\\left\[ I_3 \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F}+\\left\[ I_4 \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F} \\end{split} \[I\]ΣMF=i∑NmPi⋅\[(R PiF)TR PiF⋅E−R PiF(R PiF)T\]=i∑NmPi⋅\[(R MF+\[QMF\]R PiM)T(R MF+\[QMF\]R PiM)⋅E−(R MF+\[QMF\]R PiM)(R MF+\[QMF\]R PiM)T\]=⎩ ⎨ ⎧ mtotal⋅\[(R MF)TR MF⋅E−R MF(R MF)T\]\[I1\]ΣMF+ \[QMF\](i∑NmPi⋅\[(R PiM)TR PiM⋅E−R PiM(R PiM)T\])\[QMF\]T+\[I2\]ΣMF mtotal⋅\[(R MF)T(\[QMF\]R CoMM)⋅E−R MF(\[QMF\]R CoMM)T\]\[I3\]ΣMF+ mtotal⋅\[(\[QMF\]R CoMM)TR MF⋅E−(\[QMF\]R CoMM)(R MF)T\]\[I4\]ΣMF=\[I1\]ΣMF+\[I2\]ΣMF+\[I3\]ΣMF+\[I4\]ΣMF > 其中， \[ I 2 \] Σ M F = \[ Q M F \] ( ∑ i N m P i ⋅ \[ ( R ⃗ P i M ) T R ⃗ P i M ⋅ E − R ⃗ P i M ( R ⃗ P i M ) T \] ) \[ Q M F \] T = \[ Q M F \] \[ I \] Σ M M \[ Q M F \] T \\left\[ I_2 \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F}=\\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\left( \\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left\[ \\left( \\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^{\\mathrm{T}}\\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M}\\cdot E-\\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M}\\left( \\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^{\\mathrm{T}} \\right\]} \\right) \\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \^{\\mathrm{T}}=\\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\left\[ I \\right\] _{\\Sigma _{\\mathrm{M}}}\^{M}\\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \^{\\mathrm{T}} \[I2\]ΣMF=\[QMF\](∑iNmPi⋅\[(R PiM)TR PiM⋅E−R PiM(R PiM)T\])\[QMF\]T=\[QMF\]\[I\]ΣMM\[QMF\]T，对上式进行讨论： > > * **纯回转：** 当 R ⃗ M F = 0 \\vec{R}_{\\mathrm{M}}\^{F}=0 R MF=0时，化简为： > \[ I \] Σ M F ∣ R ⃗ M F = 0 = \[ I 2 \] Σ M F = \[ Q M F \] ( ∑ i N m P i ⋅ \[ ( R ⃗ P i M ) T R ⃗ P i M ⋅ E − R ⃗ P i M ( R ⃗ P i M ) T \] ) \[ Q M F \] T = \[ Q M F \] \[ I \] Σ M M \[ Q M F \] T \\left. \\left\[ I \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F} \\right\|_{\\vec{\\mathrm{R}}_{\\mathrm{M}}\^{F}=0}=\\left\[ I_2 \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F}=\\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\left( \\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left\[ \\left( \\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^{\\mathrm{T}}\\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M}\\cdot E-\\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M}\\left( \\vec{R}_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^{\\mathrm{T}} \\right\]} \\right) \\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \^{\\mathrm{T}}=\\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \\left\[ I \\right\] _{\\Sigma _{\\mathrm{M}}}\^{M}\\left\[ Q_{\\mathrm{M}}\^{F} \\right\] \^{\\mathrm{T}} \[I\]ΣMF R MF=0=\[I2\]ΣMF=\[QMF\](i∑NmPi⋅\[(R PiM)TR PiM⋅E−R PiM(R PiM)T\])\[QMF\]T=\[QMF\]\[I\]ΣMM\[QMF\]T > * **纯移动：** 当 R ⃗ M F ≠ 0 \\vec{R}_{\\mathrm{M}}\^{F}\\ne 0 R MF=0且 \[ Q M F \] = E \\left\[ Q_{\\mathrm{M}}\^{F} \\right\] =E \[QMF\]=E时，化简为： > \[ I \] Σ M F ∣ R ⃗ M F ≠ 0 , \[ Q M F \] = E = \[ I 1 \] Σ M F + \[ I \] Σ M M \\left. \\left\[ I \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F} \\right\|_{\\vec{\\mathrm{R}}_{\\mathrm{M}}\^{F}\\ne 0,\\left\[ Q_{\\mathrm{M}}\^{F} \\right\] =\\mathrm{E}}=\\left\[ I_1 \\right\] _{\\Sigma _{\\mathrm{M}}}\^{F}+\\left\[ I \\right\] _{\\Sigma _{\\mathrm{M}}}\^{M} \[I\]ΣMF R MF=0,\[QMF\]=E=\[I1\]ΣMF+\[I\]ΣMM > 上式也称为惯性矩阵的`平行轴定理Parallel Axis Theorem`。 > * **运动坐标系原点与质心点重合：** 当 R ⃗ C o M F = 0 \\vec{R}_{\\mathrm{CoM}}\^{F}=0 R CoMF=0时，化简为： > \[ I \] F ∣ R ⃗ C o M F = 0 = \[ I 1 \] + \[ I 2 \] \\left. \\left\[ I \\right\] \^F \\right\|_{\\vec{R}_{\\mathrm{CoM}}\^{F}=0}=\\left\[ I_1 \\right\] +\\left\[ I_2 \\right\] \[I\]F R CoMF=0=\[I1\]+\[I2
2.4 惯性矩阵的主轴定理} Principal Axis Theorem
进一步观察惯性矩阵：
I \] M = \[ ∑ i N m P i ⋅ \[ ( y P i M ) 2 + ( z P i M ) 2 \] − ∑ i N m P i ⋅ x P i M y P i M − ∑ i N m P i ⋅ ( x P i M z P i M ) − ∑ i N m P i ⋅ ( y P i M x P i M ) ∑ i N m P i ⋅ \[ ( x P i M ) 2 + ( z P i M ) 2 \] − ∑ i N m P i ⋅ ( y P i M z P i M ) − ∑ i N m P i ⋅ ( z P i M x P i M ) − ∑ i N m P i ⋅ ( z P i M y P i M ) ∑ i N m P i ⋅ \[ ( x P i M ) 2 + ( y P i M ) 2 \] \] \\left\[ I \\right\] \^M=\\left\[ \\begin{matrix} \\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left\[ \\left( y_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^2+\\left( z_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^2 \\right\]}\& -\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot x_{\\mathrm{P}_{\\mathrm{i}}}\^{M}y_{\\mathrm{P}_{\\mathrm{i}}}\^{M}}\& -\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left( x_{\\mathrm{P}_{\\mathrm{i}}}\^{M}z_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right)}\\\\ -\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left( y_{\\mathrm{P}_{\\mathrm{i}}}\^{M}x_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right)}\& \\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left\[ \\left( x_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^2+\\left( z_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^2 \\right\]}\& -\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left( y_{\\mathrm{P}_{\\mathrm{i}}}\^{M}z_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right)}\\\\ -\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left( z_{\\mathrm{P}_{\\mathrm{i}}}\^{M}x_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right)}\& -\\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left( z_{\\mathrm{P}_{\\mathrm{i}}}\^{M}y_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right)}\& \\sum_i\^N{m_{\\mathrm{P}_{\\mathrm{i}}}\\cdot \\left\[ \\left( x_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^2+\\left( y_{\\mathrm{P}_{\\mathrm{i}}}\^{M} \\right) \^2 \\right\]}\\\\ \\end{matrix} \\right\] \[I\]M= ∑iNmPi⋅\[(yPiM)2+(zPiM)2\]−∑iNmPi⋅(yPiMxPiM)−∑iNmPi⋅(zPiMxPiM)−∑iNmPi⋅xPiMyPiM∑iNmPi⋅\[(xPiM)2+(zPiM)2\]−∑iNmPi⋅(zPiMyPiM)−∑iNmPi⋅(xPiMzPiM)−∑iNmPi⋅(yPiMzPiM)∑iNmPi⋅\[(xPiM)2+(yPiM)2\] ，为`对称矩阵Symmetric Matrix`（此时默认 M M M 点与 F F F 点重合），则一定能够对角化。 等价于找到另一原点与 M M M 重合的坐标系 B B B ，使得： \[ I \] B = \[ I x x B 0 0 0 I y y B 0 0 0 I z z B \] \\left\[ I \\right\] \^B=\\left\[ \\begin{matrix} I_{\\mathrm{xx}}\^{B}\& 0\& 0\\\\ 0\& I_{\\mathrm{yy}}\^{B}\& 0\\\\ 0\& 0\& I_{\\mathrm{zz}}\^{B}\\\\ \\end{matrix} \\right\] \[I\]B= IxxB000IyyB000IzzB ，根据`矩阵对角化Matrix Diagonalizing`的原理，结合**纯回转** 推导可得： \[ I \] M = \[ Q B M \] \[ I \] B \[ Q B M \] T \\left\[ I \\right\] \^M=\\left\[ Q_{\\mathrm{B}}\^{M} \\right\] \\left\[ I \\right\] \^B\\left\[ Q_{\\mathrm{B}}\^{M} \\right\] \^{\\mathrm{T}} \[I\]M=\[QBM\]\[I\]B\[QBM\]T > 其中： > > * \[ Q B M \] \\left\[ Q_{\\mathrm{B}}\^{M} \\right\] \[QBM\] 满足 \[ i ⃗ B j ⃗ B k ⃗ B \] = \[ Q B M \] T \[ i ⃗ M j ⃗ M k ⃗ M \] \\left\[ \\begin{array}{c} \\vec{i}\^B\\\\ \\vec{j}\^B\\\\ \\vec{k}\^B\\\\ \\end{array} \\right\] =\\left\[ Q_{\\mathrm{B}}\^{M} \\right\] \^{\\mathrm{T}}\\left\[ \\begin{array}{c} \\vec{i}\^M\\\\ \\vec{j}\^M\\\\ \\vec{k}\^M\\\\ \\end{array} \\right\] i Bj Bk B =\[QBM\]T i Mj Mk M ； > * ( I x x B , I y y B , I z z B ) \\left( I_{\\mathrm{xx}}\^{B},I_{\\mathrm{yy}}\^{B},I_{\\mathrm{zz}}\^{B} \\right) (IxxB,IyyB,IzzB) 为矩阵 \[ I \] M \\left\[ I \\right\] \^M \[I\]M的`特征值Eigenvalue`； > * \[ Q B M \] \\left\[ Q_{\\mathrm{B}}\^{M} \\right\] \[QBM\] 为对应于特征值矩阵 \[ I \] B \\left\[ I \\right\] \^B \[I\]B的`特征基Standard Eigenvalue Basis`(列向量)；  