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)2wxPiF−(xOPiFyOPiF)wyPiF−(xOPiFzOPiF)wzPiF−(yOPiFxOPiF)wxPiF+(xOPiF)2+(zOPiF)2wyPiF−(yOPiFzOPiF)wzPiF−(zOPiFxOPiF)wxPiF−(zOPiFyOPiF)wyPiF+(xOPiF)2+(yOPiF)2wzPiF =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=IΣM/GFα MF+ω MF×(IΣM/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 IM。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 =QMFT I^J^K^ ，其中 Q M F T \left Q_{\\mathrm{M}}\^{F} \\right ^{\mathrm{T}} QMFT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 QMFT=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}} QMFT）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+QMFR 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} IΣMF=i∑NmPi⋅(R PiF)TR PiF⋅E−R PiF(R PiF)T=i∑NmPi⋅(R MF+\[QMFR PiM)T(R MF+QMFR PiM)⋅E−(R MF+QMFR PiM)(R MF+QMFR PiM)T]=⎩ ⎨ ⎧ mtotal⋅(R MF)TR MF⋅E−R MF(R MF)TI1ΣMF+ QMF(i∑NmPi⋅(R PiM)TR PiM⋅E−R PiM(R PiM)T)QMFT+I2ΣMF mtotal⋅(R MF)T(\[QMFR CoMM)⋅E−R MF(QMFR CoMM)T]I3ΣMF+ mtotal⋅(\[QMFR CoMM)TR MF⋅E−(QMFR CoMM)(R MF)T]I4ΣMF=I1ΣMF+I2ΣMF+I3ΣMF+I4ΣMF
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}} I2ΣMF=QMF(∑iNmPi⋅(R PiM)TR PiM⋅E−R PiM(R PiM)T)QMFT=QMFIΣMMQMFT，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}} IΣMF R MF=0=I2ΣMF=QMF(i∑NmPi⋅(R PiM)TR PiM⋅E−R PiM(R PiM)T)QMFT=QMFIΣMMQMFT- 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} IΣMF R MF=0,QMF=E=I1ΣMF+IΣMM
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] IM= ∑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 IB= 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}} IM=QBMIBQBMT
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 =QBMT 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 IM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 IB的
特征基Standard Eigenvalue Basis(列向量)；
