Lie Groups and Lie Algebras

最新推荐文章于 2022-04-14 23:56:03 发布

成风醉雅

最新推荐文章于 2022-04-14 23:56:03 发布

阅读量984

点赞数

CC 4.0 BY-SA版权

分类专栏： SLAM Math

本文链接：https://blog.youkuaiyun.com/chengfzy/article/details/81029593

SLAM 同时被 2 个专栏收录

3 篇文章

订阅专栏

Math

1 篇文章

订阅专栏

本文详细介绍了在SLAM（同时定位与地图构建）领域中常用的李群和李代数基础知识，包括特殊正交群SO(3)及其对应的李代数so(3)，以及特殊欧几里得群SE(3)和其对应的李代数se(3)。文中还探讨了指数映射和对数映射，并给出了Rodrigues公式等连接李群与李代数的重要概念。

一些在SLAM中经常用到的关于李群李代数方面的知识

Special Orthogonal Group $SO(3)$ and $\mathfrak{so}(3)$

The group $SO(3)$ forms a smooth manifold, and its tangent space (at the identity) denoted as $\mathfrak{so}(3)$ .

S O (3) s o (3) = {R \in R 3 \times 3 ∣ R R ⊺ = I, det (R) = 1} = {ϕ \in R 3, Φ = ϕ \land \in R 3 \times 3}

$\begin{aligned} SO(3) &= \left\{\mathbf{R} \in \mathbb{R}^{3\times3} \mid \mathbf{R} \mathbf{R}^\intercal = \mathbf{I}, \det(\mathbf{R}) = 1 \right\} \\ \mathfrak{so}(3) &= \left\{ \boldsymbol{\phi}\in \mathbb{R}^3, \mathbf{\Phi} = \boldsymbol{\phi}^{\wedge{}} \in \mathbb{R}^{3 \times 3} \right\} \end{aligned}$
where

ϕ \land = ⎡ ⎣ ⎢ 0 ϕ 3 - ϕ 2 - ϕ 3 0 ϕ 1 ϕ 2 - ϕ 1 0 ⎤ ⎦ ⎥

$\boldsymbol{\phi} ^\wedge = \begin{bmatrix} 0 & -\phi_3 & \phi_2 \\ \phi_3 & 0 & -\phi_1 \\ -\phi_2 & \phi_1 & 0 \end{bmatrix}$

$SO(3)$ : Rotation Matrix $R$
$\mathfrak{so}(3)$ : Angle-Axis $\boldsymbol{\phi} = \theta \mathbf{a}$
Connection: Rodrigues’s Formula

Exp Map

$\mathfrak{so}(3) \rightarrow SO(3)$

R = exp (ϕ \land) = exp (θ a \land) = cos θ I + (1 - sin θ) a a ⊺ + sin θ a \land

$\mathbf{R} = \exp(\boldsymbol{\phi}^\wedge) = \exp(\theta \mathbf{a}^\wedge) = \cos\theta \;\mathbf{I} + (1 - \sin\theta) \mathbf{a} \mathbf{a}^\intercal + \sin\theta \;\mathbf{a} ^\wedge$

Introduce $\mathrm{Exp}(\boldsymbol{\phi}) = \exp(\boldsymbol{\phi} ^\wedge)$ , then the abode equation can written as:

R = E x p (ϕ) = E x p (θ a) = cos θ I + (1 - sin θ) a a ⊺ + sin θ a \land

$\mathbf{R} = \mathrm{Exp}(\boldsymbol{\phi}) = \mathrm{Exp}(\theta \mathbf{a}) = \cos\theta \;\mathbf{I} + (1 - \sin\theta) \mathbf{a} \mathbf{a}^\intercal + \sin\theta \;\mathbf{a} ^\wedge$

Log Map

$SO(3) \rightarrow \mathfrak{so}(3)$

θ R a ϕ = arccos (t r ( R ) - 1 2) = a = θ a

$\begin{aligned} \theta &= \arccos\left( \frac{\mathtt{tr}(\mathbf{R}) - 1}{2} \right) \\ \mathbf{R} \mathbf{a} &= \mathbf{a} \\ \boldsymbol{\phi} &= \theta \mathbf{a} \end{aligned}$

BCH Formula

E x p (ϕ + Δ ϕ) \approx E x p (ϕ) E x p (J r (ϕ) Δ ϕ) = E x p (J l (ϕ) Δ ϕ) E x p (ϕ)

$\mathrm{Exp}(\boldsymbol{\phi} + \Delta \boldsymbol{\phi} ) \approx \mathrm{Exp}(\boldsymbol{\phi}) \mathrm{Exp} \big(\mathbf{J}_r(\boldsymbol{\phi}) \Delta\boldsymbol{\phi} \big) = \mathrm{Exp} \big(\mathbf{J}_l(\boldsymbol{\phi}) \Delta\boldsymbol{\phi}\big) \mathrm{Exp}(\boldsymbol{\phi})$

where

J l (ϕ) = J r (- ϕ) = sin θ θ I + (1 - sin θ θ) a a ⊺ + (1 - cos θ θ) a \land

$\mathbf{J}_l(\boldsymbol{\phi}) = \mathbf{J}_r(\mathbf{-\phi}) = \frac{\sin\theta}{\theta} \mathbf{I} + \left(1 - \frac{\sin\theta}{\theta} \right) \mathbf{a} \mathbf{a}^\intercal + \left(1 - \frac{\cos\theta}{\theta} \right) \mathbf{a}^\wedge$

L o g (E x p (ϕ) E x p (δ ϕ)) \approx L o g (E x p (δ ϕ) E x p (ϕ)) \approx ϕ + J - 1 r (ϕ) δ ϕ J - 1 l (ϕ) δ ϕ + ϕ

$\begin{aligned} \mathrm{Log}\big(\mathrm{Exp}(\boldsymbol{\phi}) \mathrm{Exp}(\delta\boldsymbol{\phi}) \big) \approx& \boldsymbol{\phi} + \mathbf{J}_r^{-1}(\boldsymbol{\phi}) \delta\boldsymbol{\phi} \\ \mathrm{Log}\big(\mathrm{Exp}(\delta\boldsymbol{\phi}) \mathrm{Exp}(\boldsymbol{\phi}) \big) \approx& \mathbf{J}_l^{-1}(\boldsymbol{\phi}) \delta\boldsymbol{\phi} + \boldsymbol{\phi}\\ \end{aligned}$

Property

R E x p (ϕ) R ⊺ = \Leftrightarrow E x p (ϕ) R = exp (R ϕ \land R ⊺) = E x p (R ϕ) R E x p (R ⊺ ϕ)

$\begin{aligned} \mathbf{R} \mathrm{Exp}(\boldsymbol{\phi}) \mathbf{R}^\intercal =& \exp(\mathbf{R} \boldsymbol{\phi}^\wedge \mathbf{R}^\intercal) = \mathrm{Exp}(\mathbf{R} \boldsymbol{\phi}) \\ \Leftrightarrow \quad \mathrm{Exp}(\boldsymbol{\phi}) \mathbf{R} =& \mathbf{R} \mathrm{Exp}(\mathbf{R}^\intercal \boldsymbol{\phi}) \end{aligned}$

Special Euclidean Group $SE(3)$ and $\mathfrak{se}(3)$

S E (3) s e (3) = {T = [R 0 ⊺ p 1] \in R 4 \times 4 ∣ R \in S O (3), p \in R 3} = {ξ = [ρ ϕ] \in R 6, ρ \in R 3, ϕ \in s o (3), ξ \land = [ϕ \land 0 ⊺ ρ 0] \in R 4 \times 4}

$\begin{aligned} SE(3) &= \left\{ \mathbf{T} = \begin{bmatrix} \mathbf{R} & \mathbf{p} \\ \mathbf{0}^\intercal & 1 \\ \end{bmatrix} \in \mathbb{R}^{4 \times 4} \mid \mathbf{R} \in SO(3), \mathbf{p} \in \mathbb{R}^3 \right\} \\ \mathfrak{se}(3) &= \left\{ \xi= \begin{bmatrix} \mathbf{\rho} \\ \boldsymbol{\phi} \end{bmatrix} \in \mathbb{R}^6, \mathbf{\rho} \in \mathbb{R}^3, \boldsymbol{\phi} \in \mathfrak{so}(3), \xi ^\wedge = \begin{bmatrix} \boldsymbol{\phi}^\wedge & \mathbf{\rho} \\ \mathbf{0}^\intercal & 0 \end{bmatrix} \in \mathbb{R}^{4 \times 4} \right\}\end{aligned}$

$SE(3)$ : Pose (Rotation $\mathbf{R}$ + Translation $\mathbf{p}$ ), Matrix
$\mathfrak{se}(3)$ : Angle Axis $\boldsymbol{\phi}$ + Translation $\mathbf{\rho}$ , Vector

Property

T - 1 = [R ⊺ 0 ⊺ - R ⊺ p 1]

$\mathbf{T} ^{-1} = \begin{bmatrix} \mathbf{R}^\intercal & -\mathbf{R}^\intercal \mathbf{p} \\ \mathbf{0}^\intercal & 1 \end{bmatrix}$

Gaussian Random Variables

R ˜ T ˜ = R E x p (Δ ϕ) = [R E x p (Δ ϕ) 0 ⊺ p + R Δ p 1]

$\begin{aligned} \widetilde{\mathbf{R}} &= \mathbf{R} \mathrm{Exp}(\Delta\boldsymbol{\phi}) \\ \widetilde{\mathbf{T}} &= \begin{bmatrix} \mathbf{R} \mathrm{Exp}(\Delta\boldsymbol{\phi}) & \mathbf{p} + \mathbf{R} \Delta\mathbf{p} \\ \mathbf{0}^\intercal & 1 \end{bmatrix} \end{aligned}$

where $\mathbf{R}$ and $\mathbf{p}$ is the noise-free rotation and translation respectively, and $\Delta\boldsymbol{\phi}$ and $\Delta\mathbf{p}$ is the perturbations (Gaussian white noise)