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一些常见的关于旋转的函数对旋转求导过程的推导

简介

主要进行了一些常见的关于旋转的函数对旋转求导过程的推导。

旋转矩阵函数对自变量求导

我们为什么要关注在李群或者李代数上叠加微小量的情况呢?因为在求导过程中,我们常见的做法是对自变量添加一个微小值来进行:

$$
f'(x) = \lim_{\Delta x\rightarrow0}\frac{f(x+\Delta x)}{\Delta x}
$$

对于旋转矩阵 $SO(3)$ 我们不能这么做,因为李群对加法不封闭,因此两个旋转矩阵相加不一定是旋转矩阵,但是利用李代数,根据上面两个方向的 BCH 近似不难看出我们有两种思路进行求导,分别是:

  • 用李代数(旋转向量)来表示姿态,然后利用李代数加法叠加微小量并对该微小量进行求导
  • 用李群(旋转矩阵)表示姿态,然后左/右乘上一个扰动,然后对该扰动求导,即左扰动模型和右扰动模型

旋转点的求导推导

下面举例,假设我们对空间一个点 $\boldsymbol{p}$ 使用旋转矩阵 $\boldsymbol{R}$ 进行旋转得到 $\boldsymbol{Rp}$,我们想要计算旋转后的点的坐标对于旋转的导数,记为:

$$
\frac{\partial \boldsymbol{Rp}}{\partial \boldsymbol{R}}
$$
  • 李代数求导

首先将旋转矩阵(李群 $SO(3)$)转换为旋转向量(李代数 $\boldsymbol{\mathfrak{so}}(3)$),并对旋转向量求导:

$$
\begin{aligned}
    \frac{\partial \boldsymbol{Rp}}{\partial \boldsymbol{R}} = \frac{\partial(\exp{(\boldsymbol{\phi}^\wedge})\boldsymbol{p})}{\partial\boldsymbol{\phi}} &= \lim_{\delta \boldsymbol{\phi\rightarrow 0}}\frac{\exp{(\boldsymbol{(\phi + \delta\phi)}^\wedge})\boldsymbol{p} - \exp{(\boldsymbol{\phi}^\wedge})\boldsymbol{p}}{\delta\phi} \\
    \mathrm{BCH 近似}&\approx \lim_{\delta \boldsymbol{\phi\rightarrow 0}}\frac{\exp{\boldsymbol{((J_l\mathrm{\delta}\phi)^\wedge)}}\exp{(\boldsymbol{\phi}^\wedge)}\boldsymbol{p} - \exp{(\boldsymbol{\phi}^\wedge})\boldsymbol{p}}{\delta\phi} \\
    \mathrm{泰勒展开并去除高阶项}&\approx \lim_{\delta \boldsymbol{\phi\rightarrow 0}}\frac{(\boldsymbol{I + (J_l\mathrm{\delta}\phi)^\wedge)}\exp{(\boldsymbol{\phi}^\wedge)}\boldsymbol{p} - \exp{(\boldsymbol{\phi}^\wedge})\boldsymbol{p}}{\delta\phi} \\
    \mathrm{简化}&= \lim_{\delta \boldsymbol{\phi\rightarrow 0}}\frac{\boldsymbol{(J_l\mathrm{\delta}\phi)^\wedge}\exp{(\boldsymbol{\phi}^\wedge)}\boldsymbol{p}}{\delta\phi} \\
    \mathrm{利用a^\wedge b = -b^\wedge a 性质}&= \lim_{\delta \boldsymbol{\phi\rightarrow 0}}\frac{-(\exp{(\boldsymbol{\phi}^\wedge)}\boldsymbol{p})^\wedge\boldsymbol{(J_l\mathrm{\delta}\phi)^\wedge}}{\delta\phi} \\
    \mathrm{化简} &= \boldsymbol{-(Rp)^\wedge J_l} \\
\end{aligned}
$$

最后我们得到以下结果:

$$
\frac{\partial \boldsymbol{Rp}}{\partial \boldsymbol{R}} = \frac{\partial(\exp{(\boldsymbol{\phi}^\wedge})\boldsymbol{p})}{\partial\boldsymbol{\phi}} = \boldsymbol{-(Rp)^\wedge J_l(\phi)}
$$

其中涉及到 $\boldsymbol{J_l(\phi)}$, 因此相对而言还是比较复杂。

  • 左扰动模型

在旋转矩阵上左乘一个扰动,并对扰动进行求导。

$$
\begin{aligned}
    \frac{\partial(\boldsymbol{R}\boldsymbol{p})}{\partial\boldsymbol{\psi}} 
    &= \lim_{\psi\rightarrow0}\frac{\exp{(\boldsymbol{\psi}^\wedge{})}\exp{(\boldsymbol{\phi}^\wedge{})}\boldsymbol{p}
                                    - \exp{(\boldsymbol{\phi}^\wedge{})}\boldsymbol{p}}{\boldsymbol{\psi}} \\
    \mathrm{泰勒展开并去除高阶项}&\approx \lim_{\psi\rightarrow0}\frac{(\boldsymbol{I+\boldsymbol{\psi}^\wedge{}})\exp{(\boldsymbol{\phi}^\wedge{})}\boldsymbol{p} 
                                    - \exp{(\boldsymbol{\phi}^\wedge{})}\boldsymbol{p}}{\boldsymbol{\psi}} \\
    \mathrm{简化}&= \lim_{\psi\rightarrow0}\frac{\boldsymbol{\psi}^\wedge{}\boldsymbol{R}\boldsymbol{p}}{\boldsymbol{\psi}} \\
    \mathrm{利用a^\wedge b = -b^\wedge a 性质}&= \lim_{\psi\rightarrow0}\frac{-(\boldsymbol{Rp})^\wedge{}\boldsymbol{\psi}}{\boldsymbol{\psi}} \\
    &= -\boldsymbol{(Rp)}^\wedge{}
\end{aligned}
$$

不难看出来,利用左扰动模型计算的导数比使用李代数直接求导省去了一个 $\boldsymbol{J_l(\phi)}$ 的计算,因此更为使用。

  • 右扰动模型

在旋转矩阵上右乘一个扰动,并对扰动进行求导。

$$
\begin{aligned}
    \frac{\partial(\boldsymbol{R}\boldsymbol{p})}{\partial\boldsymbol{\psi}} 
    &= \lim_{\psi\rightarrow0}\frac{\exp{(\boldsymbol{\phi}^\wedge{})}\exp{(\boldsymbol{\psi}^\wedge{})}\boldsymbol{p}
                                    - \exp{(\boldsymbol{\phi}^\wedge{})}\boldsymbol{p}}{\boldsymbol{\psi}} \\
    \mathrm{泰勒展开并去除高阶项}&\approx \lim_{\psi\rightarrow0}\frac{\exp{(\boldsymbol{\phi}^\wedge{})}(\boldsymbol{I+\boldsymbol{\psi}^\wedge{}})\boldsymbol{p}
                                    - \exp{(\boldsymbol{\phi}^\wedge{})}\boldsymbol{p}}{\boldsymbol{\psi}} \\
    \mathrm{简化}&= \lim_{\psi\rightarrow0}\frac{\boldsymbol{R}\boldsymbol{\psi}^\wedge{}\boldsymbol{p}}{\boldsymbol{\psi}} \\
    \mathrm{利用a^\wedge b = -b^\wedge a 性质}&= \lim_{\psi\rightarrow0}\frac{\boldsymbol{R}(-\boldsymbol{p}^\wedge{}\boldsymbol{\psi})}{\boldsymbol{\psi}} \\
    &= -\boldsymbol{Rp}^\wedge{}
\end{aligned}
$$

不难看出来,相比于左扰动的模型中计算 $\boldsymbol{Rp}$ 的反对称矩阵,右扰动模型计算的是 $\boldsymbol{p}$ 的反对称矩阵,因此有细微的区别,使用时注意区分。

旋转连乘的求导的推导

除了旋转点以外,我们还经常需要对两个旋转叠加的结果 $\boldsymbol{R_1R_2}$对其中一个旋转进行求导,即:

$$
\frac{\mathrm{d}\ln{\boldsymbol{(R_1R_2)}^\vee}}{\boldsymbol{R_2}}\\
\frac{\mathrm{d}\ln{\boldsymbol{(R_1R_2)}^\vee}}{\boldsymbol{R_1}}
$$

首先,推导对 $\boldsymbol{R_2}$ 求导过程。

  • 右扰动模型

同样地我们转化为李代数进行求导:

$$
\begin{aligned}
    \frac{\mathrm{d}\ln{\boldsymbol{(R_1R_2)}^\vee}}{\boldsymbol{R_2}} &= \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\ln{(\boldsymbol{R_1R_2}\exp{(\boldsymbol{\phi}^\wedge}))^\vee}-\ln{(\boldsymbol{R_1R_2})^\vee}}{\boldsymbol{\phi}} \\
    \mathrm{BCH 近似}&\approx \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\ln{(\boldsymbol{R_1R_2})^\vee} + \boldsymbol{J}_r^{-1}\boldsymbol{\phi}-\ln{(\boldsymbol{R_1R_2})^\vee}}{\boldsymbol{\phi}} \\
    \mathrm{化简}&= \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\boldsymbol{J}_r^{-1}\boldsymbol{\phi}}{\boldsymbol{\phi}} \\
    \mathrm{分子分母抵消} &= \boldsymbol{J}_r^{-1} = \boldsymbol{J}_r^{-1}(\ln{(\boldsymbol{R_1R_2})^\vee})
\end{aligned}
$$
  • 左扰动模型

过程大同小异。

$$
\begin{aligned}
    \frac{\mathrm{d}\ln{\boldsymbol{(R_1R_2)}^\vee}}{\boldsymbol{R_2}} &= \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\ln{(\boldsymbol{R_1}\exp{(\boldsymbol{\phi}^\wedge})\boldsymbol{R_2})^\vee}-\ln{(\boldsymbol{R_1R_2})^\vee}}{\boldsymbol{\phi}} \\
    \mathrm{SO(3) 的伴随性质}&= \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\ln{(\boldsymbol{R_1}\boldsymbol{R_2}\exp{((\boldsymbol{R_2^\mathrm{T}\phi})^\wedge)})^\vee}-\ln{(\boldsymbol{R_1R_2})^\vee}}{\boldsymbol{\phi}} \\
    \mathrm{BCH 近似}&\approx \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\ln{(\boldsymbol{R_1}\boldsymbol{R_2})^\vee}+\boldsymbol{J^\mathrm{-1}_rR_\mathrm{2}^\mathrm{T}\phi}-\ln{(\boldsymbol{R_1R_2})^\vee}}{\boldsymbol{\phi}} \\
    \mathrm{化简}&= \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\boldsymbol{J^\mathrm{-1}_rR_\mathrm{2}^\mathrm{T}\phi}}{\boldsymbol{\phi}} \\
    \mathrm{分子分母抵消} &= \boldsymbol{J^\mathrm{-1}_rR_\mathrm{2}^\mathrm{T}} = \boldsymbol{J}_r^{-1}(\ln{(\boldsymbol{R_1R_2})^\vee})\boldsymbol{R_\mathrm{2}^\mathrm{T}}
\end{aligned}
$$

$\boldsymbol{R_1}$ 求导。

  • 右扰动模型

过程如下:

$$
\begin{aligned}
    \frac{\mathrm{d}\ln{\boldsymbol{(R_1R_2)}^\vee}}{\boldsymbol{R_1}} &= \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\ln{(\boldsymbol{R_1}\exp{(\boldsymbol{\phi}^\wedge})\boldsymbol{R_2})^\vee}-\ln{(\boldsymbol{R_1R_2})^\vee}}{\boldsymbol{\phi}} \\
    \mathrm{SO(3) 的伴随性质}&= \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\ln{(\boldsymbol{R_1}\boldsymbol{R_2}\exp{((\boldsymbol{R_2^\mathrm{T}\phi})^\wedge)})^\vee}-\ln{(\boldsymbol{R_1R_2})^\vee}}{\boldsymbol{\phi}} \\
    \mathrm{BCH 近似}&= \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\ln{(\boldsymbol{R_1}\boldsymbol{R_2})^\vee}+\boldsymbol{J^\mathrm{-1}_rR_\mathrm{2}^\mathrm{T}\phi}-\ln{(\boldsymbol{R_1R_2})^\vee}}{\boldsymbol{\phi}} \\
    \mathrm{化简}&= \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\boldsymbol{J^\mathrm{-1}_rR_\mathrm{2}^\mathrm{T}\phi}}{\boldsymbol{\phi}} \\
    \mathrm{分子分母抵消} &= \boldsymbol{J^\mathrm{-1}_rR_\mathrm{2}^\mathrm{T}} = \boldsymbol{J}_r^{-1}(\ln{(\boldsymbol{R_1R_2})^\vee})\boldsymbol{R_\mathrm{2}^\mathrm{T}}
\end{aligned}
$$
  • 左扰动模型

过程如下:

$$
\begin{aligned}
    \frac{\mathrm{d}\ln{\boldsymbol{(R_1R_2)}^\vee}}{\boldsymbol{R_1}} &= \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\ln{(\exp{(\boldsymbol{\phi}^\wedge})\boldsymbol{R_1R_2})^\vee}-\ln{(\boldsymbol{R_1R_2})^\vee}}{\boldsymbol{\phi}} \\
    \mathrm{BCH 近似}&\approx \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\ln{(\boldsymbol{R_1R_2})^\vee} + \boldsymbol{J}_l^{-1}\boldsymbol{\phi}-\ln{(\boldsymbol{R_1R_2})^\vee}}{\boldsymbol{\phi}} \\
    \mathrm{化简}&= \lim_{\boldsymbol{\phi}\rightarrow0}\frac{\boldsymbol{J}_l^{-1}\boldsymbol{\phi}}{\boldsymbol{\phi}} \\
    \mathrm{分子分母抵消} &= \boldsymbol{J}_l^{-1} = \boldsymbol{J}_l^{-1}(\ln{(\boldsymbol{R_1R_2})^\vee})
\end{aligned}
$$

可以发现,无论是对 $\boldsymbol{R}_1$ 还是 $\boldsymbol{R}_2$ 使用左扰动或者右扰动求导,我们总是想办法使用 BCH 或者 伴随性质将 $\boldsymbol{R}_1\boldsymbol{R}_2$ 凑到一起与第二项相减,最后抵消扰动获得结果,过程大同小异。

逆旋转点的求导

下面进行对逆旋转点的函数的求导,即:

$$
\frac{\mathrm{d}(\boldsymbol{R^\mathrm{-1}p})}{\mathrm{d}\boldsymbol{R}}
$$
  • 左扰动模型

过程如下:

$$
\begin{aligned}
    \frac{\mathrm{d}(\boldsymbol{R^\mathrm{-1}p})}{\mathrm{d}\boldsymbol{R}} &= \lim_{\boldsymbol{\psi}\rightarrow 0} \frac{(\exp(\boldsymbol{\psi}^\wedge)\exp(\boldsymbol{\phi}^\wedge))^{-1}\boldsymbol{p}-\exp(\boldsymbol{\phi}^\wedge)^{-1}\boldsymbol{p}}{\boldsymbol{\psi}} \\
    逆矩阵的性质 &= \lim_{\boldsymbol{\psi}\rightarrow 0} \frac{\exp(\boldsymbol{\phi}^\wedge)^{-1}\exp(\boldsymbol{\psi}^\wedge)^{-1}\boldsymbol{p}-\exp(\boldsymbol{\phi}^\wedge)^{-1}\boldsymbol{p}}{\boldsymbol{\psi}} \\
    指数映射的性质 &= \lim_{\boldsymbol{\psi}\rightarrow 0} \frac{\exp(\boldsymbol{\phi}^\wedge)^{-1}\exp(\boldsymbol{-\psi}^\wedge)\boldsymbol{p}-\exp(\boldsymbol{\phi}^\wedge)^{-1}\boldsymbol{p}}{\boldsymbol{\psi}} \\
    \mathrm{泰勒展开并去除高阶项} &\approx \lim_{\boldsymbol{\psi}\rightarrow 0} \frac{\boldsymbol{R}^{-1}(\boldsymbol{I+(-\psi)}^\wedge)\boldsymbol{p}-\boldsymbol{R}^{-1}\boldsymbol{p}}{\boldsymbol{\psi}} \\
    化简 &= \lim_{\boldsymbol{\psi}\rightarrow 0} \frac{\boldsymbol{R}^{-1}\boldsymbol{p} - \boldsymbol{R}^{-1}\psi^\wedge\boldsymbol{p}-\boldsymbol{R}^{-1}\boldsymbol{p}}{\boldsymbol{\psi}} \\
    &= \lim_{\boldsymbol{\psi}\rightarrow 0} \frac{- \boldsymbol{R}^{-1}\psi^\wedge\boldsymbol{p}}{\boldsymbol{\psi}} \\
    \mathrm{利用a^\wedge b = -b^\wedge a 性质} &= \lim_{\boldsymbol{\psi}\rightarrow 0} \frac{- \boldsymbol{R}^{-1}(-\boldsymbol{p^\wedge})\psi}{\boldsymbol{\psi}} \\
    分子分母抵消 &= \boldsymbol{R}^{-1}\boldsymbol{p^\wedge}
\end{aligned}
$$
  • 右扰动模型

过程如下:

$$
\begin{aligned}
\frac{\mathrm{d} \left( \boldsymbol{R}^{-1} \boldsymbol{p} \right) }{ \mathrm{d} \boldsymbol{R}} &= \lim _{\phi \rightarrow 0} \frac{ \left( \boldsymbol{R} \exp \left( \phi^{\wedge} \right) \right)^{-1} \boldsymbol{p} - \boldsymbol{R}^{-1} \boldsymbol{p} }{\phi } \\
    逆矩阵的性质 &= \lim _{\phi \rightarrow 0} \frac{ \left( \exp \left( \phi^{\wedge}\right)\right)^{-1} \boldsymbol{R} ^{-1} \boldsymbol{p} - \boldsymbol{R}^{-1} \boldsymbol{p} }{\phi} \\
    指数映射的性质 &= \lim _{\phi \rightarrow 0} \frac{ \exp (-\phi)^{\wedge} \boldsymbol{R}^{-1} \boldsymbol{p} - \boldsymbol{R}^{-1} \boldsymbol{p} }{\phi} \\
    \mathrm{泰勒展开并去除高阶项} &\approx \lim _{\phi \rightarrow 0} \frac{ \left( \boldsymbol{I} + (-\phi)^{\wedge}\right) \boldsymbol{R}^{-1} \boldsymbol{p} - \boldsymbol{R}^{-1} \boldsymbol{p} }{\phi} \\
    化简 &= \lim _{\phi \rightarrow 0} \frac{ (-\phi)^{\wedge} \boldsymbol{R}^{-1} \boldsymbol{p} }{\phi} \\
    反对称算子性质 &= \lim _{\phi \rightarrow 0} \frac{ \left( \boldsymbol{R}^{-1} \boldsymbol{p} \right)^{\wedge} \phi }{ \phi } \\
    分子分母抵消&= \left( \boldsymbol{R}^{-1} \boldsymbol{p} \right)^{\wedge}
\end{aligned}
$$

逆旋转函数的求导

求解以下函数的导数:

$$
\frac{\mathrm{d}\ln(\boldsymbol{R_\mathrm{1}R_\mathrm{2}^\mathrm{-1}})^\vee}{\mathrm{d}\boldsymbol{R}_2}
$$
  • 左扰动模型

过程如下:

$$
\begin{aligned}
    \frac{\mathrm{d}\ln(\boldsymbol{R_\mathrm{1}R_\mathrm{2}^\mathrm{-1}})^\vee}{\mathrm{d}\boldsymbol{R}_2} &= 
    \lim_{\boldsymbol{\psi}\rightarrow 0}\frac{\ln{(\boldsymbol{R}_1(\exp{(\boldsymbol{\psi}^\wedge)}\boldsymbol{R}_2)^{-1})^\vee}- \ln{(\boldsymbol{R}_1\boldsymbol{R}_2^{-1})^\vee}}{\boldsymbol{\psi}} \\
    逆矩阵的性质 &= \lim_{\boldsymbol{\psi}\rightarrow 0}\frac{\ln{(\boldsymbol{R}_1\boldsymbol{R}_2^{-1}\exp{(\boldsymbol{\psi}^\wedge)}^{-1})^\vee}- \ln{(\boldsymbol{R}_1\boldsymbol{R}_2^{-1})^\vee}}{\boldsymbol{\psi}} \\
    指数映射性质 &= \lim_{\boldsymbol{\psi}\rightarrow 0}\frac{\ln{(\boldsymbol{R}_1\boldsymbol{R}_2^{-1}\exp{(\boldsymbol{-\psi}^\wedge)})^\vee}- \ln{(\boldsymbol{R}_1\boldsymbol{R}_2^{-1})^\vee}}{\boldsymbol{\psi}} \\
    \mathrm{BCH 近似} &\approx \lim_{\boldsymbol{\psi}\rightarrow 0}\frac{\ln{(\boldsymbol{R}_1\boldsymbol{R}_2^{-1})^\vee} + \boldsymbol{J_r^{-1}}(-\boldsymbol{\psi})- \ln{(\boldsymbol{R}_1\boldsymbol{R}_2^{-1})^\vee}}{\boldsymbol{\psi}} \\
    化简 &= \lim_{\boldsymbol{\psi}\rightarrow 0}\frac{- \boldsymbol{J_r^{-1}}\boldsymbol{\psi}}{\boldsymbol{\psi}} \\
    分子分母抵消 &= - \boldsymbol{J_r^{-1}}\boldsymbol{\psi} = - \boldsymbol{J_r^{-1}}(\ln{(\boldsymbol{R_1R_2^{-1}})})
\end{aligned}
$$
  • 右扰动模型

过程如下:

$$
\begin{aligned}
\frac{\mathrm{d} \ln \left(\boldsymbol{R}{1} \boldsymbol{R}{2}^{-1}\right)^{\vee} } {\mathrm{d} \boldsymbol{R}{2}} &= \lim_{\phi \rightarrow 0} \frac {\ln \left( \boldsymbol{R}_1 \left( \boldsymbol{R}_2 \exp \left( \phi^{\wedge}\right) \right)^{-1}\right)^{\vee} - \ln \left( \boldsymbol{R}_1 \boldsymbol{R}2^{-1} \right)^{\vee} } {\phi} \\
逆矩阵的性质 &= \lim_{\phi \rightarrow 0} \frac{ \ln \left( \boldsymbol{R}_1 \left( \exp \left( \phi^\wedge \right) \right)^{-1} \boldsymbol{R}_2^{-1} \right)^{\vee} - \ln \left( \boldsymbol{R}_1 \boldsymbol{R}2^{-1} \right)^{\vee} }{\phi} \\
指数映射性质&= \lim_{\phi \rightarrow 0} \frac{ \ln \left( \boldsymbol{R}_1 \exp \left( -\phi^\wedge \right) \boldsymbol{R}_2^{-1} \right)^{\vee} - \ln \left( \boldsymbol{R}_1 \boldsymbol{R}2^{-1} \right)^{\vee} }{\phi} \\
\mathrm{SO(3) 的伴随性质} &= \lim_{\phi \rightarrow 0} \frac { \ln \left( \boldsymbol{R}_1 \boldsymbol{R}_2^{-1} \exp \left( \boldsymbol{R}_2 \left( -\phi \right) \right)^{\wedge} \right)^{\vee} - \ln \left( \boldsymbol{R}_1 \boldsymbol{R}_2^{-1} \right)^{\vee} }{\phi} \\
\mathrm{BCH 近似}&\approx \lim_{ \phi \rightarrow 0 } \frac { \ln \left( \boldsymbol{R}_1 \boldsymbol{R}_2^{-1} \right)^{\vee} - \boldsymbol{J}_r ^{-1} \left( \ln( \boldsymbol{R}_1 \boldsymbol{R}_2^{-1} ) \right) \boldsymbol{R}_2 \phi - \ln \left( \boldsymbol{R}_1 \boldsymbol{R}_2^{-1} \right)^{\vee} }{ \phi } \\
\mathrm{化简} &= -\boldsymbol{J}_r^{-1} \left( \ln( \boldsymbol{R}_1 \boldsymbol{R}_2^{-1} )^{\vee } \right) \boldsymbol{R}_2 
\end{aligned}
$$
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