> ## Documentation Index > Fetch the complete documentation index at: https://aegean.ai/llms.txt > Use this file to discover all available pages before exploring further. # Lie cheatsheet > Lie groups and Lie algebras for rigid-body motion, with their elements, dimensions, and meanings. A 4×4 homogeneous transformation matrix represents a rigid-body configuration. A rotation matrix $R \in SO(3)$ represents orientation. A six-dimensional twist describes spatial velocity. A six-dimensional wrench represents spatial force. The table uses the rigid-body motion notation of Lynch and Park (Kevin M. Lynch, 2017). | Space | Elements | Dimension | Represents | | - | - | - | - | | $SO(2)$ | $2 \times 2$ matrices $R$ with $R^T R = I$, $\det R = 1$ | 1 | Planar orientation | | $\mathfrak{so}(2)$ | $2 \times 2$ skew-symmetric matrices $\begin{bmatrix} 0 & -\omega \\ \omega & 0 \end{bmatrix}$ | 1 | Planar angular velocity | | $SE(2)$ | $3 \times 3$ matrices $\begin{bmatrix} R & \mathbf{p} \\ 0 & 1 \end{bmatrix}$, $R \in SO(2)$, $\mathbf{p} \in \mathbb{R}^2$ | 3 | Planar configuration $(x, y, \theta)$ | | $\mathfrak{se}(2)$ | $3 \times 3$ matrices $\begin{bmatrix} [\omega] & v \\ 0 & 0 \end{bmatrix}$, $[\omega] \in \mathfrak{so}(2)$, $v \in \mathbb{R}^2$ | 3 | Planar twist | | $SO(3)$ | $3 \times 3$ matrices $R$ with $R^T R = I$, $\det R = 1$ | 3 | Orientation in 3D | | $\mathfrak{so}(3)$ | $3 \times 3$ skew-symmetric matrices $[\omega]$ | 3 | Angular velocity, exponential coordinates $[\hat{\omega}]\theta$ | | $SE(3)$ | $4 \times 4$ matrices $T = \begin{bmatrix} R & \mathbf{p} \\ 0 & 1 \end{bmatrix}$, $R \in SO(3)$, $\mathbf{p} \in \mathbb{R}^3$ | 6 | Configuration (pose) in 3D | | $\mathfrak{se}(3)$ | $4 \times 4$ matrices $[V] = \begin{bmatrix} [\omega] & v \\ 0 & 0 \end{bmatrix}$, $[\omega] \in \mathfrak{so}(3)$, $v \in \mathbb{R}^3$ | 6 | Twist, screw axis $[\mathcal{S}]\theta$ | | $\mathbb{R}^3$ | Vectors | 3 | Position $\mathbf{p}$, angular velocity $\omega$, exponential coordinates $\hat{\omega}\theta$ | | $\mathbb{R}^6$ | Vectors | 6 | Twist $V = (\omega, v)$, wrench $F = (\tau, f)$ | | $S^3$ | Unit quaternions $q$ with $\lVert q \rVert = 1$ | 3 | Orientation; $q$ and $-q$ give the same rotation | $SO(n)$ and $SE(n)$ are Lie groups. They are the curved spaces of configurations. Their Lie algebras, $\mathfrak{so}(n)$ and $\mathfrak{se}(n)$, are flat tangent spaces at the identity that contain velocities. The matrix exponential maps a velocity to a configuration: $\exp: \mathfrak{so}(3) \to SO(3)$ and $\exp: \mathfrak{se}(3) \to SE(3)$. The matrix logarithm maps a configuration to a velocity that produces it in unit time: $\log: SO(3) \to \mathfrak{so}(3)$ and $\log: SE(3) \to \mathfrak{se}(3)$. The exponential is many-to-one, so a rotation has many logarithms. In the plane, the angles $\theta$ and $\theta + 2\pi$ give the same rotation. Choosing the principal angle in $(-\pi, \pi]$ makes the planar logarithm a function. On that branch, it inverts the exponential. In three dimensions, restricting the rotation angle to $[0, \pi]$ does not make the logarithm unique. A rotation by exactly $\pi$ has two logarithms because the axes $\hat{\omega}$ and $-\hat{\omega}$ give the same rotation. The logarithm is a single-valued inverse for angles below $\pi$. A rotation by $\pi$ requires a sign convention for the axis. The $[\cdot]$ operator maps a vector in $\mathbb{R}^3$ or $\mathbb{R}^6$ to its matrix form in $\mathfrak{so}(3)$ or $\mathfrak{se}(3)$. ## Rotations and rigid-body motions side by side Lynch and Park summarize chapter 3 in four tables (Kevin M. Lynch, 2017). Each row pairs a fact about rotation with its rigid-body counterpart. Table comparing rotations and rigid-body motions. R in SO(3) is a 3 by 3 matrix with R transpose R equal to I and determinant 1; T in SE(3) is a 4 by 4 matrix built from R and p. Inverses: R inverse equals R transpose; T inverse has R transpose and minus R transpose p. Change of frame: R_ab R_bc equals R_ac and T_ab T_bc equals T_ac. Table continued. Rotating a frame by R equals Rot of omega hat and theta, premultiplying for the fixed axis and postmultiplying for the body axis; displacing a frame by T, likewise. A unit rotation axis omega hat with norm 1, against a unit screw axis S with either norm of omega equal to 1, or omega zero and norm of v equal to 1. A screw axis with point q, direction s hat and finite pitch h. Angular velocity omega equals omega hat theta dot; twist V equals S theta dot. Table continued. The skew-symmetric matrix of omega in so(3) and its identities; the matrix form of a twist in se(3). R dot R inverse equals omega s and R inverse R dot equals omega b; T dot T inverse equals V s and T inverse T dot equals V b. The adjoint matrix with R, the bracket of p times R and R, with its inverse and product identities. Table continued. Change of frame: omega_a equals R_ab omega_b; S_a and V_a equal the adjoint of T_ab times S_b and V_b. Exponential coordinates omega hat theta in R3 and S theta in R6. Exp from so(3) to SO(3) by Rodrigues' formula, and from se(3) to SE(3). Log for each, with algorithms in sections 3.2.3.3 and 3.3.3.2. Moment change of frame m_a equals R_ab m_b; wrench change of frame F_a equals the transpose of the adjoint of T_ba times F_b. ## References * Kevin M. Lynch, F. (2017). *Modern Robotics*. *** Found an error on this page? [Report an issue](https://github.com/aegean-ai/eaia-feedback/issues/new?template=page-issue.yml\&page=/aiml-common/lectures/motion-algebras/lie-cheatsheet). This documentation is built and hosted on [Mintlify](https://mintlify.com), a developer documentation platform.