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

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.

• Actually, we don't even all know what SE stands for (apart from StackExchange); I'll take myself as counterexample. My guess: Special Euclidean? May 25, 2016 at 12:55

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