the physical significance of the Lie Algebra of SE(3)

Question

as we all know, the Lie group of $SE(3)$ can be written in the form of $4\times4$ matrix, say $$ \begin{pmatrix} R & t\\ 0 & 1 \end{pmatrix},\tag{1} $$ and its Lie Algebra, denoted as $se(3)$, can be composed by 6 generators $$ \begin{aligned} G_1&=\begin{pmatrix}0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} & G_2&=\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} & G_3&=\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \end{pmatrix}\\ \\ G_4&=\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} & G_5&=\begin{pmatrix}0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} & G_6&=\begin{pmatrix}0 & -1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix}, \end{aligned}\tag{2} $$ so its Lie Algebra can be represented by multiples of the generators in the form of $4\times4$ matrix, $$ \begin{pmatrix} w_x & u\\ 0 & 0 \end{pmatrix}.\tag{3} $$ We can use the coordinates of rotation axis to build $w_x$, skew symmetric matrix, and the coordinates of translation vector to build $u$, so by exponential map, we can get homogeneous matrix in the form of (3) which we can use in the displacement of a frame. I am confused that why we can use the coordinates of rotation axis and translation vector to build $se(3)$ by multiples of the six generators, or what is the link among rotation axis ,translation vector and the six generators? What is the physical significance of the $se(3)$? I hope that you can help me by explain the physical significance of $se(2)$ or $so(2)$ first to me. Thanks for anyone's help.

Answer 1

Physically, SE(3) (the Special Euclidean Group in 3 dimensions) is the group of simultaneous rotations and translations for a vector. It is heavily used in robotics and general kinematics. However, we rarely refer to it as SE(3) except in the most pedantic of texts. This is because it represents something that doesn't really gain anything by moving into abstract algebra. A common example that uses the special euclidean group would be this:

Let us assume that I have a vector $\vec{p} = <x,y,z>$ with respect to some coordinate frame $F_i$. For fun, let's say that this is the coordinate frame attached to the palm of a robot hand. Physically, this point may represent the center of the robot's tool, but reckoned with respect to the palm of the hand. I'll often know this quite nicely, but it's useless to me because I want the robot to affect something in the world. In other words, I know where I want that tool to be in the world frame, not in the palm frame, which moves with the robot. To fix this, I need to do two things: first, I need to rotate the vector so that it is aligned with the world frame, and second I have to translate it so that it is reckoned from the origin of the world frame, rather than the palm. To do this, I use a transformation matrix $T_p^W \in SE(3)$* to make a new vector $\vec{p}_w = T_p^W \vec{p}$. Now, the tool position is known in the world frame and I'm ready to start doing path planning.

Long story short, SE(3) moves vectors from one frame to another. These frames may be rotated and translated with respect to each other. It is heavily used in robotics and other places where multiple coordinate frames matter.

To answer the other questions: the SO(n) groups are just the groups of rotation matrices (ignoring translation), and SE(2) is the same as SE(3) but in 2 dimensions. SE(2) and SO(2) are actually used, since it is common for mechanisms to only be able to move in a plane, rather than full 3D space, so it greatly simplifies the math to restrict things to 2D.

One important thing to note, though, is that the group representation of mechanisms is almost always way too abstract to be needed. The SE(n) groups were among the first studied, so we prefer to talk about things like "rotation matrices" and "translation matrices" rather than "group elements of SE(n)", even though they act the same way on the mechanisms we tend to use them on!

*technically, $T$ is in a representation of SE(3), but I'll take what I can get