假设只发生旋转，下标e表示地球系，下标b表示机体系。

反对称矩阵的表示方法 a × b = [ a ] × b a \times b = [a]_{\times}b a×b=[a]×​b 其中： [ a ] × = [ 0 − a z a y a z 0 − a x − a y a x 0 ] [a]_{\times}=\left[ \begin{matrix} 0 & -a_z & a_y \\ a_z & 0 & -a_x \\ -a_y & a_x & 0 \end{matrix} \right] [a]×​=⎣⎡​0az​−ay​​−az​0ax​​ay​−ax​0​⎦⎤​

公理 d r e ⃗ d t = ω e ⃗ × r e ⃗ \frac{d\vec{r_e}}{dt}=\vec{\omega_e} \times \vec{r_e} dtdre​ ​​=ωe​ ​×re​ ​

推导 将旋转矩阵看成3个地球系表达的列向量的组合 d R b e d t = d [ b 1 e ⃗ b 2 e ⃗ b 3 e ⃗ ] d t = [ ω e ⃗ × b 1 e ⃗ ω e ⃗ × b 2 e ⃗ ω e ⃗ × b 3 e ⃗ ] = [ ( R b e ω b ⃗ ) × b 1 e ⃗ ( R b e ω b ⃗ ) × b 2 e ⃗ ( R b e ω b ⃗ ) × b 3 e ⃗ ] = [ ( R b e ω b ⃗ ) × ( R b e e 1 ⃗ ) ( R b e ω b ⃗ ) × ( R b e e 2 ⃗ ) ( R b e ω b ⃗ ) × ( R b e e 3 ⃗ ) ] \frac{d{R_b^e}}{dt}=\frac{d[\vec{b_1^e}\quad\vec{b_2^e}\quad\vec{b_3^e}]}{dt}=[\vec{\omega_e} \times \vec{b_1^e} \quad \vec{\omega_e} \times \vec{b_2^e} \quad \vec{\omega_e} \times \vec{b_3^e}]\\=[(R_b^e\vec{\omega_b}) \times \vec{b_1^e} \quad (R_b^e\vec{\omega_b}) \times \vec{b_2^e} \quad (R_b^e\vec{\omega_b}) \times \vec{b_3^e}]\\=[(R_b^e\vec{\omega_b}) \times (R_b^e\vec{e_1}) \quad (R_b^e\vec{\omega_b}) \times (R_b^e\vec{e_2}) \quad (R_b^e\vec{\omega_b}) \times (R_b^e\vec{e_3})] dtdRbe​​=dtd[b1e​ ​b2e​ ​b3e​ ​]​=[ωe​ ​×b1e​ ​ωe​ ​×b2e​ ​ωe​ ​×b3e​ ​]=[(Rbe​ωb​ ​)×b1e​ ​(Rbe​ωb​ ​)×b2e​ ​(Rbe​ωb​ ​)×b3e​ ​]=[(Rbe​ωb​ ​)×(Rbe​e1​ ​)(Rbe​ωb​ ​)×(Rbe​e2​ ​)(Rbe​ωb​ ​)×(Rbe​e3​ ​)]

其中： e 1 ⃗ = [ 1 0 0 ] T \vec{e_1}=[1\quad0\quad0]^T e1​ ​=[100]T e 2 ⃗ = [ 0 1 0 ] T \vec{e_2}=[0\quad1\quad0]^T e2​ ​=[010]T e 3 ⃗ = [ 0 0 1 ] T \vec{e_3}=[0\quad0\quad1]^T e3​ ​=[001]T 对于旋转矩阵 R R R和向量叉乘，有以下性质（本文不证明了，有兴趣的可以自己推导） ( R a ⃗ ) × ( R b ⃗ ) = R ( a ⃗ × b ⃗ ) (R\vec{a}) \times (R\vec{b})=R(\vec{a} \times \vec{b}) (Ra )×(Rb )=R(a ×b )

应用该性质得到： d R b e d t = [ R b e ( ω b ⃗ × e 1 ⃗ ) R b e ( ω b ⃗ × e 2 ⃗ ) R b e ( ω b ⃗ × e 3 ⃗ ) ] = R b e [ ω b ⃗ × e 1 ⃗ ω b ⃗ × e 2 ⃗ ω b ⃗ × e 3 ⃗ ] = R b e [ [ ω b ⃗ ] × e 1 ⃗ [ ω b ⃗ ] × e 2 ⃗ [ ω b ⃗ ] × e 3 ⃗ ] = R b e [ ω b ⃗ ] × \frac{d{R_b^e}}{dt}=[R_b^e(\vec{\omega_b} \times \vec{e_1}) \quad R_b^e(\vec{\omega_b} \times \vec{e_2}) \quad R_b^e(\vec{\omega_b} \times \vec{e_3})]\\=R_b^e[\vec{\omega_b} \times \vec{e_1} \quad \vec{\omega_b} \times \vec{e_2} \quad \vec{\omega_b} \times \vec{e_3}]\\=R_b^e[[\vec{\omega_b}]_{\times} \vec{e_1} \quad [\vec{\omega_b}]_{\times} \vec{e_2} \quad [\vec{\omega_b}]_{\times} \vec{e_3}]\\=R_b^e[\vec{\omega_b}]_{\times} dtdRbe​​=[Rbe​(ωb​ ​×e1​ ​)Rbe​(ωb​ ​×e2​ ​)Rbe​(ωb​ ​×e3​ ​)]=Rbe​[ωb​ ​×e1​ ​ωb​ ​×e2​ ​ωb​ ​×e3​ ​]=Rbe​[[ωb​ ​]×​e1​ ​[ωb​ ​]×​e2​ ​[ωb​ ​]×​e3​ ​]=Rbe​[ωb​ ​]×​

综上所述： d R b e d t = R b e [ ω b ⃗ ] × \frac{d{R_b^e}}{dt}=R_b^e[\vec{\omega_b}]_{\times} dtdRbe​​=Rbe​[ωb​ ​]×​