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% Math% Some specific variables % extrinsic baseline vector % camera frame % Euler angles % Motion matrix % Map % element of SO(3) % quaternion % rotation matrix % matrix % translation vector % rotation vector
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为什么？
在机器人学中，我们经常需要描述物体的位置和姿态。位置好说，用欧几里得空间 \(\mathbb{R}^3\) 的向量就行，可以直接加减。但是姿态（旋转）很特殊。
如果你用旋转矩阵 \(R\) 来描述姿态，它必须满足 \(R^\top R = I\) 且 \(\det(R)=1\)。这意味着旋转矩阵不是在空间中随意分布的，而是受到严格约束，分布在一个弯曲的表面上。这种光滑的弯曲表面，数学上称为流形（Manifold）。
想象一只蚂蚁在一个球面上爬行。球面就是“流形”。蚂蚁不能直接沿直线飞到球内部，它必须沿着球画弧线。我们在机器人中遇到的旋转 \(SO(3)\) 和刚体变换 \(SE(3)\)，本质上就是这种“弯曲的空间”。 传统的卡尔曼滤波（KF）假设状态量服从高斯分布（\(x \sim \mathcal{N}(\mu, \Sigma)\)），这暗示状态空间是平直的线性空间。如果在弯曲的旋转流形上强行用线性空间的加法（比如两个旋转矩阵直接相加），结果通常不再是一个合法的旋转矩阵。因此，我们需要李群理论来建立一套严谨的微积分体系，以处理旋转和平移的不确定性、导数和积分。
具体来说在状态估计中，构造损失函数和优化问题 \[ r = f(\mathbf X) \] where \(r\) 是损失函数，\(f\) 为非线性函数，\(\mathbf X\) 为状态。 \[ \mathbf X^* = \min_\mathbf X r \] 一般采用迭代方法求解，GD, GN, LM等。每一步进行线性化，找到前进的方向（GD）。 需要线性化，求导。
若 \(\mathbf{X}\) 在向量空间内，求导用矩阵求导就可。 若 \(\mathbf{X} : \mathbf{R}_{2\times2}\) 旋转矩阵（流形），如何求导？此时： \[ r = f(\mathbf{R}_{2\times2}) = f(\mathbf R, t) \] Vector space -> Manifold space 引入Lie group 用于对这种特殊空间进行求导。
李群与李代数的基本定义 (The Lie Group and Lie Algebra)
李群 (The Lie Group, \(\mathcal{M}\))
李群 \(\mathcal{G}\) 是一个集合，它同时具备两个性质：
群（Group）的代数性质：有闭合的运算（比如矩阵乘法）、有单位元、有逆元、满足结合律。
流形（Manifold）的几何性质：它是光滑的，局部看起来像线性空间。
常见的李群包括： - \(S^1\): 单位复数（2D 旋转）。
\(SO(3)\): 3D 旋转矩阵。
\(SE(3)\): 3D 刚体变换（旋转 + 平移）。
什么是李群（Lie Group）？ 定义：李群 \(\mathcal{G}\) 是一个光滑流形，其元素满足群公理。
它必须满足四个公理： 1. 封闭性 (Closure): \(\mathcal{X}\circ \mathcal{Y}\in \mathcal{G}\)
单位元 (Identity): \(\mathcal{E}\circ \mathcal{X}= \mathcal{X}\circ \mathcal{E}= \mathcal{X}\)
逆元 (Inverse): \(\mathcal{X}^{-1} \circ \mathcal{X}= \mathcal{E}\)
结合律 (Associativity): \((\mathcal{X}\circ \mathcal{Y}) \circ \mathcal{Z}= \mathcal{X}\circ (\mathcal{Y}\circ \mathcal{Z})\)
In many groups of interest, the operation \(\circ\) is non-commutative. For example, the rotations in 3D are non-commutative, the rotations in 2D are commutative.
Group actions
Lie groups come with the power to transform elements of other sets, producing \({\it}{e.g.}\) rotations, translations, scaling, and combinations of them. These are extensively used in robotics, both in 2D and 3D.
Given a Lie group \(\mathcal{M}\) and a set \(\mathcal{V}\), we note \(\mathcal{X}\cdot v\) the \({\it}{action}\) of \(\mathcal{X}\in\mathcal{M}\) on \(v\in\mathcal{V}\), \[ \begin{align} \cdot~:~\mathcal{M}\times\mathcal{V}\to\mathcal{V}~;~ (\mathcal{X},v)\mapsto\mathcal{X}\cdot v ~. \end{align} \] For \(\cdot\) to be a group action, it must satisfy the axioms, \[ \begin{align} \textrm{Identity} &:& \mathcal{E}\cdot v &= v \\ \textrm{Compatibility} &:& (\mathcal{X}\circ\mathcal{Y})\cdot v &= \mathcal{X}\cdot(\mathcal{Y}\cdot v) ~. \end{align} \] Lie groups were formerly known as Continuous Transformation groups in 19 century.
Common examples are the groups of rotation matrices \(\textit{SO}(n)\), the group of unit quaternions, and the groups of rigid motion \(\textit{SE}(n)\). Their respective actions on vectors satisfy \[ \begin{align} \textit{SO}(n) &:\textrm{rotation matrix } & {\bf R}\cdot{\bf x}&\triangleq{\bf R}{\bf x}\\ \textit{SE}(n) &:\textrm{Euclidean matrix } & {\bf H}\cdot{\bf x}&\triangleq{\bf R}{\bf x}+ {\bf t}\\ S^1 &:\textrm{unit complex } & {\bf z}\cdot{\bf x}&\triangleq{\bf z}\,{\bf x}\\ S^3 &:\textrm{unit quaternion } & {\bf q}\cdot{\bf x}&\triangleq{\bf q}\,{\bf x}\,{\bf q}^* \end{align} \]
切空间与李代数 (Tangent Space and Lie Algebra)
既然流形是弯曲的，为了做计算（比如求导、定义协方差），我们需要一个平直的线性空间来近似它。
切空间 (Tangent Space): 在流形上任意一点 \(\mathcal{X}\)，都切着一个平面，记为 \(T_{\mathcal{X}}\mathcal{M}\)。
李代数 (Lie Algebra, \(\mathfrak{m}\)): 特指在单位元 \(\mathcal{E}\) 处的切空间，记为 \(T_{\mathcal{E}}\mathcal{M}\)，或者简写为 \(\mathfrak{m}\)。
\[ \textrm{Lie Algebra} : \quad\quad \mathfrak{m} \triangleq T_{\mathcal{E}}\mathcal{M} \]
Any Lie algebra, the vector space is endowed with a non-associative product called the Lie bracket. Here we don’t talk about this.
对于 \(m\) 维的流形，其切空间同构于欧几里得空间 \(\mathbb{R}^m\)。 \[ {\it}{e.g.}\quad {\mathfrak {so}}(3) \simeq {\mathbb R}^3 \] 作符号约定以区分李代数元素和向量坐标：
笛卡尔向量空间 \(\mathbb{R}^m\): 元素用 \(\boldsymbol{\tau}\) 表示。这是我们存储数据、计算协方差矩阵的地方。
李代数 \(\mathfrak{m}\): 元素用 \(\boldsymbol{\tau}^\wedge\) 表示（通常是反对称矩阵或虚数）。
在李群理论中，我们需要在向量空间 \(\mathbb{R}^m\) 和李代数 \(\mathfrak{m}\) 之间转换。 它们之间通过 Hat \((\cdot)^\wedge\) 和 Vee \((\cdot)^\vee\) 算子进行转换。
请写出 Hat \((\cdot)^\wedge\) 和 Vee \((\cdot)^\vee\) 映射的作用。 \[ \begin{align} \textrm{Hat} &: \quad \mathbb{R}^m \to \mathfrak{m}; \quad \boldsymbol{\tau} \to \boldsymbol{\tau}^\wedge = \sum_{i=1}^m \tau_i E_i \\ \textrm{Vee} &: \quad \mathfrak{m} \to \mathbb{R}^m; \quad \boldsymbol{\tau}^\wedge \to (\boldsymbol{\tau}^\wedge)^\vee = \boldsymbol{\tau} = \sum_{i=1}^m \tau_i {\bf e}_i \,. \end{align} \] The element \(\boldsymbol{\tau}^\wedge\) of the Lie algebra have non-trivial structures (skew-symmetric matrices, imaginary numbers, pure quaternions…). They can be expressed as linear combinations of some base elements \(E_i\), where \(E_i\) are called the generators of \(\mathfrak m\) (They are derivatives of \(\mathcal{X}\) around the origin in the \(i\)-th direction).
\({\bf e}_i\) are the vectors of the base of \({\mathbb R}^m\) (we have \({\bf e}_i^\wedge = E_i\)). This means that \(\mathfrak{m}\) is isomorphic to the vector space \({\mathbb R}^m\).
Hat \((\cdot)^\wedge\) 和 Vee \((\cdot)^\vee\) 映射是 互逆线性映射 (mutually inverse linear maps)，或 同构，isomorphisms。
什么是同构？
从此我们能得出什么结论？ \[ \begin{align} \mathfrak m &\simeq {\mathbb R}^m \\ \boldsymbol{\tau}^\wedge &\simeq \boldsymbol{\tau} \\ \boldsymbol{\tau} &\in {\mathbb R}^m \\ \boldsymbol{\tau}^\wedge &\in \mathfrak m \end{align} \]
延伸
以 \(SO(3)\) 为例： - \(\boldsymbol{\omega} \in \mathbb{R}^3\) 是一个 3 维向量（角速度）。
- \([\boldsymbol{\omega}]_\times \in \mathfrak{so}(3)\) 是一个 \(3 \times 3\) 的反对称矩阵。
这里的 Hat 算子就是把向量变成反对称矩阵： \[ \boldsymbol{\omega}^\wedge = [\boldsymbol{\omega}]_\times = \begin{bmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{bmatrix} \] 这种矩阵形式是后续计算指数映射的基础。
Each element of the group has an equivalent in the Lie algebra. This relation is so profound that (nearly) all operations in the group, which is curved and nonlinear, have an exact equivalent in the Lie algebra, which is a linear vector space.
In engineering, vectors \(\boldsymbol{\tau} \in {\mathbb R}^m\) are handier than their isomorphic \(\boldsymbol{\tau}^\wedge \in \mathfrak m\), since they can be stacked in larger state vectors, and more importantly, manipulated with linear algebra using matrix operators.
不完全理解的东西
李代数里的元素不是向量，而是能作用在群上的无穷小生成元。 李代数元素是：一个“变化方向 + 变化规则”， 也就是说：它不是“在哪里”，而是“如果动，会怎么动”。它不是空间里的位移向量，状态量，指向某个点的箭头。
在 SO(3) 里： \(\omega^\wedge \in \mathfrak{so}(3)\) 它的意义是：如果我沿着这个方向流动 infinitesimal 时间 \(dt\)， 我会产生一个旋转 形式上： \[R(t) = \exp(t\,\omega^\wedge)\]
| \(\omega\) | 参数 | | \(\omega^\wedge\) | 生成旋转的算子 | | \(\exp(t\omega^\wedge)\) | 流（flow） | | Lie bracket | 两种“动法”的不交换性 |
切空间的例子
Tangent space of \(S^1\)
Consider the velocity of a point: Differentiate \(z^* \cdot z = 1\) w.r.t time: \[ \begin{align} z^*z + z^*\dot z &= 0 \\ z^* \dot z &= -(z^* \dot z)^* \leftarrow \textrm{Imaginary}\\ z^* \dot z &= i\omega \in i{\mathbb R}\leftarrow \textrm{Imaginary} \end{align} \] \[ \begin{align} \textrm{Lie algebra} &:& \omega^\wedge &= i \cdot \omega \in i{\mathbb R}\\ \textrm{Cartesian} &:& \omega &\in {\mathbb R}\\ \textrm{Isomorphism} &\downarrow& \\ \textrm{Hat} &:& \omega^\wedge &= i \cdot \omega \\ \textrm{Vee} &:& \omega &= -i \cdot \omega^\vee ~. \end{align} \]
Tangent space of SO(3)
Differentiate \({\bf R}^\top \cdot {\bf R}= {\bf I}\) w.r.t. time: \[ \begin{align} {\bf R}^\top {\bf R}+ {\bf R}^\top \dot {\bf R}&= 0 \\ {\bf R}^\top \dot {\bf R}&= -({\bf R}^\top \dot {\bf R})^\top \leftarrow \textrm{Skew-symmetric}\\ {\bf R}^\top \dot {\bf R}&= \begin{bsmallmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{bsmallmatrix} \in {\mathfrak {so}}(3) \leftarrow \textrm{Skew-symmetric} \end{align} \] Lie algebra when \({\bf R}={\bf I}\) \[ \begin{align} \textrm{Lie algebra} &:& \dot{\bf R}= \left[\boldsymbol \omega\right]_\times &\triangleq \begin{bmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{bmatrix} \in {\mathfrak {so}}(3) \\ && &= \omega_x \begin{bsmallmatrix}0&0&0 \\ 0&0&-1 \\ 0&1&0 \end{bsmallmatrix} + \omega_y \begin{bsmallmatrix}0&0&1 \\ 0&0&0 \\ -1&0&0 \end{bsmallmatrix} + \omega_z \begin{bsmallmatrix}0&-1&0 \\ 1&0&0 \\ 0&0&0 \end{bsmallmatrix} \\ \textrm{Cartesian} \,\,\,\, {\mathbb R}^3&:& \boldsymbol \omega &= \begin{bmatrix} \omega_x & \omega_y& \omega_z \end{bmatrix}^\top \in {\mathbb R}^3 \\ &&& = \omega_x \begin{bsmallmatrix}1&0&0\end{bsmallmatrix}^\top + \omega_y \begin{bsmallmatrix}0&1&0\end{bsmallmatrix}^\top + \omega_z \begin{bsmallmatrix}0&0&1\end{bsmallmatrix}^\top \\ \textrm{Isomorphsim} &:& {\mathfrak {so}}(3) &\simeq {\mathbb R}^3 \\ \textrm{Hat} &:& \boldsymbol \omega&\mapsto\boldsymbol \omega^\wedge=\left[\boldsymbol \omega\right]_\times \\ \textrm{Vee} &:& \left[\boldsymbol \omega\right]_\times& \mapsto \left[\boldsymbol \omega\right]_\times^\vee = \boldsymbol \omega ~. \end{align} \]
指数映射与对数映射 (Exp and Log Maps)
如何将平直切空间里的微小增量应用到弯曲的流形上？这需要“卷绕”操作。
指数映射 (Exponential Map): 将切空间 \(\mathfrak{m}\) 中的切向量映射回流形 \(\mathcal{M}\)。直观上，它是沿着切向量方向在流形上走“大圆弧”（测地线 Geodesic\(^\dagger\)）。
对数映射 (Logarithmic Map): 指数映射的逆运算，将流形上的点映射回切空间。
\(^\dagger\)Geodesic: the greatest arc possible in the manifold.
Here, we define the tangent increment \(\boldsymbol{\tau} \triangleq {\bf v}t \in {\mathbb R}^m\) as velocity per time, so that we have \(\boldsymbol{\tau}^\wedge = {\bf v}^\wedge t \in \mathfrak{m}\), a point in the Lie algebra. Then the exponential map, and its inverse logarithmic map can be written as, \[ \begin{align} \exp: \mathfrak m \to \mathcal{M}; \quad \boldsymbol{\tau} &\mapsto \mathcal X = \exp(\boldsymbol{\tau}^\wedge) \\ \log: \mathcal{M} \to \mathfrak m; \quad \mathcal X &\mapsto \boldsymbol{\tau}^\wedge =\log(\mathcal{X}). \end{align} \] 为了方便工程使用，可定义大写的 Exp 和 Log，直接作用于 Cartesian 向量空间上的向量 \(\mathbb{R}^m\)： \[ \begin{align} \text{Exp}&: \mathbb{R}^m \to \mathcal{M} \quad &; \quad \boldsymbol{\tau} \mapsto \mathcal{X} &= \text{Exp}(\boldsymbol{\tau}) \triangleq \exp(\boldsymbol{\tau}^\wedge) \\ \text{Log}&: \mathcal{M} \to \mathbb{R}^m \quad &; \quad \mathcal{X}\mapsto \boldsymbol{\tau} &= \text{Log}(\mathcal{X}) \triangleq \log(\mathcal{X})^\vee \\ \end{align} \] where \(\boldsymbol{\tau} \in {\mathbb R}^m \simeq \mathcal{T}_{\mathcal{E}}\mathcal{M};\mathcal{X}\in\mathcal{M}\).
对于矩阵李群，指数映射 \(\exp(\boldsymbol{\tau}^\wedge)\) 的泰勒展开式是什么？ \[ \exp(\boldsymbol{\tau}^\wedge) = \mathcal{E} + \boldsymbol{\tau}^\wedge + \frac{1}{2!}(\boldsymbol{\tau}^\wedge)^2 + \frac{1}{3!}(\boldsymbol{\tau}^\wedge)^3 + \cdots \] 这是标量指数函数 \(e^x = \sum \frac{x^n}{n!}\) 的矩阵推广。
回忆一下切平面与流形的关系图：
广义加减法算子 (Plus and Minus Operators)
为了避免混淆普通的向量加法和流形上的运算，定义了 \(\oplus\) 和 \(\ominus\)。
Plus and minus allow us to introduce increments between elements of a curved manifold and express them in its flat tangent vector space.
因为矩阵乘法不满足交换律，所以扰动是左乘还是右乘很重要。这里约定默认使用“右乘”扰动，对应“局部坐标系”。
右加法 (Right Plus, \(\oplus\))
给流形上的状态 \(\mathcal{X}\) 施加一个局部的微小扰动 \(\boldsymbol{\tau}\)，得 \(\mathcal{Y} = \mathcal{X} \oplus \boldsymbol{\tau}\)： \[ \text{right-}\oplus:\quad \mathcal{Y}= \mathcal{X}\oplus^\mathcal{X}\!\boldsymbol{\tau} \triangleq \mathcal{X}\circ \text{Exp}(^\mathcal{X}\!\boldsymbol{\tau}) \in \mathcal{M}. \] 这意味着：先在局部切空间 \(\mathcal{X}\) 里定义一个向量 \(\boldsymbol{\tau}\)，通过 \(\text{Exp}\) 变成群元素（比如旋转矩阵），然后右乘到原状态上。
右减法 (Right Minus, \(\ominus\))
计算两个流形元素 \(\mathcal{Y}\) 和 \(\mathcal{X}\) 之间的差异，\(\boldsymbol{\tau} = \mathcal{Y} \ominus \mathcal{X}\)（结果是一个切空间向量）： \[ \text{right-}\ominus:\quad ^\mathcal{X}\!\boldsymbol{\tau} = \mathcal{Y}\ominus \mathcal{X}\triangleq \text{Log}(\mathcal{X}^{-1} \circ \mathcal{Y}) \in \mathcal{T}_\mathcal{X}\mathcal{M}. \]
为什么是 \(\mathcal{X}^{-1} \circ \mathcal{Y}\)？ 想象方程 \(\mathcal{Y}= \mathcal{X}\circ \boldsymbol{\delta}\)。我们想求差异 \(\boldsymbol{\delta}\)。 等式两边左乘 \(\mathcal{X}^{-1}\)，得到 \(\mathcal{X}^{-1} \circ \mathcal{Y}= \boldsymbol{\delta}\)。 这个 \(\boldsymbol{\delta}\) 还在群里，我们要把它变成向量，所以套一个 \(\text{Log}\)。
\(\text{Exp}(^\mathcal{X}\!\boldsymbol{\tau})\) appears at the right hand side of the composition, \(^\mathcal{X}\!\boldsymbol{\tau}\) belongs to the tangent space at \(\mathcal{X}\): we say by convention, \(^\mathcal{X}\!\boldsymbol{\tau}\) is expressed in the local frame at \(\mathcal{X}\).
左加法 (Left Plus, \(\oplus\))
给全局坐标系（世界坐标） 中的扰动 \(^{\mathcal{E}}\!\boldsymbol{\tau}\)，作用到状态 \(\mathcal{X}\) 上： \[ \text{left-}\oplus:\quad \mathcal{Y}= ^\mathcal{E}\!\boldsymbol{\tau} \oplus\mathcal{X}\triangleq \text{Exp}(^\mathcal{E}\!\boldsymbol{\tau}) \circ \mathcal{X}\in \mathcal{M}. \] 扰动不是附着在 \(\mathcal{X}\) 上的，而是在群单位元 \(\mathcal{E}\) 的切空间里定义的全局扰动，然后左乘，把整个状态一起推走。
左减法 (Left Minus, \(\ominus\))
计算两个状态之间的全局差异： \[ \text{left-}\ominus:\quad ^\mathcal{E}\!\boldsymbol{\tau} = \mathcal{Y}\ominus \mathcal{X}\triangleq \text{Log}(\mathcal{Y}\circ \mathcal{X}^{-1} ) \in \mathcal{T}_\mathcal{E}\mathcal{M}. \] 为什么左减的结果在 \(\mathcal{T}_{\mathcal{E}}\mathcal{M}\)，而不是 \(\mathcal{T}_{\mathcal{X}}\mathcal{M}\)？ 因为左减测量的是世界坐标里的群差，而不是以 \(\mathcal{X}\) 为基点的局部误差。
- \(\mathcal{T}_\mathcal{E}\mathcal{M}\)： 全局坐标、群代数、固定参考系
- \(\mathcal{T}_\mathcal{X}\mathcal{M}\)： 随状态移动的局部坐标
- 左扰动：世界在推你
- 右扰动：你在自己坐标里动
- 右加减：误差随状态走 → 切空间在 \(\mathcal{X}\)
- 左加减：误差定义在群本身 → 切空间在 \(\mathcal{E}\)
\(^\mathcal{E}\!\boldsymbol{\tau} \oplus\mathcal{X}\) 和 \(\mathcal{X}\oplus^\mathcal{X}\!\boldsymbol{\tau}\) 是什么关系？有什么意义？ \[ ^\mathcal{E}\!\boldsymbol{\tau} \oplus\mathcal{X}= \mathcal{X}\oplus^\mathcal{X}\!\boldsymbol{\tau} \] determines a relation between the local and global tangent elements.
问： \(\text{Exp}(^\mathcal{E}\!\boldsymbol{\tau})\mathcal{X}=\,?\) \[ \text{Exp}(^\mathcal{E}\!\boldsymbol{\tau})\mathcal{X}= \mathcal{X}\,\text{Exp}(^\mathcal{X}\!\boldsymbol{\tau}) \] \(^\mathcal{E}\!\boldsymbol{\tau}^\wedge = \,?\) \[ ^\mathcal{E}\!\boldsymbol{\tau}^\wedge = \mathcal{X}^\mathcal{X}\!\boldsymbol{\tau}^\wedge\mathcal{X}^{-1} \] \(\exp(^\mathcal{E}\!\boldsymbol{\tau}^\wedge)=\,?\) \[ \exp(^\mathcal{E}\!\boldsymbol{\tau}^\wedge)= \mathcal{X}\exp(^\mathcal{X}\!\boldsymbol{\tau}^\wedge)\mathcal{X}^{-1} = \exp(\mathcal{X}^\mathcal{X}\!\boldsymbol{\tau}^\wedge\mathcal{X}^{-1}) \]
伴随 (The Adjoint, \(\text{Ad}\))
我们在局部坐标系（Body frame）定义了扰动，有时候需要把它转换到全局坐标系（World frame），或者反过来。线性代数里我们用旋转矩阵旋转向量，在李群里，这个任务由伴随（Adjoint）完成。
伴随 \(\text{Ad}_{\mathcal{X}}\) 是一个线性变换，它将局部切空间的向量映射到全局切空间（或者反之，取决于定义方向，这里定义为将局部系下的 \(\boldsymbol{\tau}\) 变换为在原点 \(\mathcal{E}\) 处的切向量）。 定义： \[ \text{Ad}_{\mathcal{X}} : \mathfrak m \to \mathfrak m; \quad \boldsymbol{\tau}^\wedge \mapsto \text{Ad}_\mathcal{X}\triangleq \mathcal{X}\boldsymbol{\tau}^\wedge\mathcal{X}^{-1}, \] so that, \[ ^\mathcal{E}\!\boldsymbol{\tau}^\wedge = \text{Ad}_\mathcal{X}(^\mathcal{X}\!\boldsymbol{\tau}^\wedge). \] This defines the adjoint action of the group on its own Lie algebra. Adjoint 有两个性质： \[ \begin{align} \text{Linear} &: \quad \text{Ad}_\mathcal{X}(a\boldsymbol{\tau}^\wedge+b\boldsymbol{\sigma}^\wedge) = a\text{Ad}_\mathcal{X}(\boldsymbol{\tau}^\wedge) + b\text{Ad}_\mathcal{X}(\boldsymbol{\sigma}^\wedge) \\ \text{Homomorphism} &: \quad \text{Ad}_\mathcal{X}(\text{Ad}_\mathcal{Y}(\boldsymbol{\tau}^\wedge)) = \text{Ad}_{\mathcal{X}\mathcal{Y}}(\boldsymbol{\tau}^\wedge) \end{align} \] 公式关系： \[ \text{Exp}(\text{Ad}_{\mathcal{X}} \boldsymbol{\tau}) = \mathcal{X} \circ \text{Exp}(\boldsymbol{\tau}) \circ \mathcal{X}^{-1} \] Since \(\text{Ad}_\mathcal{X}\) is linear, we can found an equivalent matrix operator \(\mathbf{Ad}_{\mathcal{X}}\) that maps the Cartesian tangent vectors \(^\mathcal{E}\!\boldsymbol{\tau}\) and \(^\mathcal{X}\!\boldsymbol{\tau}\)： \[ \begin{align} \mathbf{Ad}_{\mathcal{X}} : {\mathbb R}^m &\to {\mathbb R}^m; \quad ^\mathcal{X}\!\boldsymbol{\tau} \mapsto \,^\mathcal{E}\!\boldsymbol{\tau} = \mathbf{Ad}_\mathcal{X}\,^\mathcal{X}\!\boldsymbol{\tau}\\ ^\mathcal{E}\!\boldsymbol{\tau} &\simeq \,^\mathcal{E}\!\boldsymbol{\tau}^\wedge; \quad ^\mathcal{X}\!\boldsymbol{\tau} \simeq \,^\mathcal{X}\!\boldsymbol{\tau}^\wedge \end{align} \] 它是一个 \(m \times m\) 的矩阵。 This can be computed by applying \(^\wedge\) to \(\text{Ad}_{\mathcal{X}}\), \[ \mathbf{Ad}_\mathcal{X}\,\boldsymbol{\tau} = (\mathcal{X}\boldsymbol{\tau}^\wedge\mathcal{X}^{-1})^\wedge \] 除了和 Adjoint 一样拥有线性同态，伴随矩阵 \(\mathbf{Ad}_{\mathcal{X}}\) 还有哪些性质？ \[ \begin{align} \mathcal{X}\, \oplus\boldsymbol{\tau} &= (\mathbf{Ad}_\mathcal{X}\,\boldsymbol{\tau})\,\oplus\mathcal{X}\\ \mathbf{Ad}_{\mathcal{X}^{-1}} &= \mathbf{Ad}_{\mathcal{X}}^{-1} \\ \mathbf{Ad}_{\mathcal{X}\mathcal{Y}} &= \mathbf{Ad}_\mathcal{X}\mathbf{Ad}_\mathcal{Y}. \end{align} \] In these two \(\mathbf{Ad}_{\mathcal{X}^{-1}} = \mathbf{Ad}_{\mathcal{X}}^{-1}\), \(\mathbf{Ad}_{\mathcal{X}\mathcal{Y}} = \mathbf{Ad}_\mathcal{X}\mathbf{Ad}_\mathcal{Y}\) equations, the left parts are usually cheaper to computer than the right ones. We will use the adjoint matrix often as a way to linearly transform vectors of the tangent space at \(\mathcal{X}\) onto vectors of the tangent space at the origin, with \(^\mathcal{E}\!\boldsymbol{\tau} = \mathbf{Ad}_\mathcal{X}\, ^\mathcal{X}\!\boldsymbol{\tau}.\)