Visual slam Lecture 3 -- Lie groups and Lie Algebras

0. Content

1. What is a group

Why introduce groups ？

Because when calculating the derivative of rotation matrix or transformation matrix , Addition no longer holds , So finding the derivative of a matrix requires introducing other methods , So the group is introduced .

“ Group , Ring , Domain ” They are all things in abstract algebra

Because addition is a commutative group on integers , therefore 2+3=3+2=5
for example ： Operations on addition , Univariate is 0,a+(-a)=0
In multiplication , Univariate is 1,$a\ast \frac{1}{a}=1$

The rotation matrix is reversible , Is an orthogonal matrix , And matrix multiplication to form a group .
If not in groups , Basically, there is no need to discuss .
General $n\times n$ The invertible matrix of is a general linear group GL(n)

Basic knowledge supplement ：《 Abstract algebra 》,《 Differential geometry 》

2. Lie groups and Lie algebras

With continuous （ smooth ） A group of properties is called a lie group （Lie Group）.
Basic knowledge supplement ：《 Differential geometry 》

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Definition of manifold （ I understand it ）： A manifold is a space . Manifold is a space formed by mapping low dimensional data to high dimensions . Data in high-dimensional space will produce dimensional redundancy , But in fact, these data can be uniquely represented only by using lower dimensions . And the manifold learning I've heard before is ： Suppose the data is a low dimensional manifold uniformly sampled in a high-dimensional Euclidean space , Manifold learning is to recover low dimensional manifold structure from high dimensional sampled data , And find low dimensional manifolds in high dimensional space , And find the corresponding embedding mapping , To achieve dimension reduction or data visualization . He is looking for the essence of things from the observed phenomena , The internal law of data generation has been found .

The following pictures are from Reference blog

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SO(3) and SE(3) Only well-defined multiplication , No addition , So it is difficult to reach the limit of retrograde , Derivation and other operations .（ It was said before that the group was introduced in order to find the derivative of the rotation matrix ？ Why can't we ask here ？）

First of all SO(3)( Rotation matrix ) To introduce Lie algebra .

First, for the rotation matrix , It is orthogonal. , Derivation of time on both sides , After finishing, we get an antisymmetric matrix
$$\dot{R}(t)R(t{)}^T$$
a^ Represents from vector to antisymmetric matrix , and

Denotes that the vector is represented by an antisymmetric matrix .

Can be regarded as right R(t) After derivation , One more on the left $\varphi (t)$^

Yes R(t) First order Taylor expansion , stay $t_0$ near $\varphi$ unchanged , Into differential equations , Bring in the initial condition solution （ There are some details that have not been discussed carefully ）

Any given vector $\varphi$, Can find a corresponding relationship ,
$$e^{{\hat{\varphi }}_{0}t}=R(t)$$
among $\varphi$ It's lie algebra , The correspondence is exponential mapping ,$R$ It's Li Qun .
The following is a strict definition of Lie algebra ：

Given the structure, we can check whether it is a lie algebra .

[,] It's a lie bracket operation .
SO(3) It's a matrix ,so(3) It's a vector ( In fact, matrix and vector correspond one by one , Fine , But this is more natural ), Lie brackets are operations that act on vectors .

SE(3) The lie algebra of is 6 Dimension vector ,$\rho$ Represents translation ,$\varphi$ For rotation , And the top tip ^ No longer represent a symmetric matrix , But still retain the notation . Just put the rotating part $\varphi$ Do antisymmetry , The translation part remains unchanged , from (6,1)->(4,4)

3. Exponential mapping and logarithmic mapping

Taylor expansion

Expand by infinite series （ See high number 18 speak P244）
$$e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}$$
$$\sin x=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{(2n+1)!}$$
$$\cos x=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n}}{(2n)!}$$
among 3 The expansions converge to $-\infty <x<+\infty$
Add the following properties to deal with higher powers

You can deduce Rodriguez formula .

Then explain Lie algebra so(3) The physical meaning of is actually rotation vector （ Rodrigues formula is rotation vector to rotation matrix , On the left is the rotation matrix $R$）. conversely , Given the rotation matrix, the rotation vector can be obtained , You can use logarithmic mapping or the following transformation relationship ：

about SE(3)

Where the translation vector $\rho$ And the translation in the transformation matrix $t$ A coefficient matrix is missing , That is Jacobian matrix , It's related to the angle , The rotating part is just a so(3).

This part summarizes

Li Qun –> Exponential mapping –> Lie algebra
Lie algebra –> Logarithmic mapping –> Li Qun

4. Derivation and perturbation model

Or derivation , Because lie groups have no addition , Therefore, the derivation is required to start from the perspective of Lie algebra , Add a small quantity to lie algebra $\delta \varphi$ Then it is transformed into Lie group by exponential operation , But the question is whether addition in Lie algebra is equivalent to multiplication of Lie groups . It is obviously true in scalar ,a,b It is obviously true when it is scalar $e^a\ast e^b=e^{a+b}$, But it doesn't hold when the power is a matrix .

introduce BCH The formula ：

Matrix two exp And then take ln Expansion is a formula about lie brackets , Quite complicated , After some approximation ：

When one of the two matrices is a small quantity , Gu slightly makes the following approximation in higher order ：

When multiplying left by a small quantity , The coefficient is the left Jacobian matrix ; When multiplying left by right by a small amount , The coefficient is the right Jacobian matrix , It is often compared with Zuo Ya .

So when adding small quantities on Lie algebra , Equivalent to the upper left of the Group （ Right ） Multiply by a quantity with Jacobi .

SE(3) Than SO(3) complex , In the top right corner of the $Q_r$ Very complicated .

There are two ways to find derivatives ：

1. Add a small quantity to lie algebra , Find the rate of change
2. Liqun is superior to a small quantity , Find the rate of change relative to lie algebra

Derivative model ：

Reverse use BCH The formula , Then Taylor expands , The quadratic term is 0 了 , So Taylor expansion only retains the first two .

Disturbance model , Left （ Right ） Multiply the group corresponding to a disturbance

SE(3) Perturbation model ： Left multiply perturbation

5. Sophus practice