Robotics
SE(3)
Definition
SE(3) is the special Euclidean group of three-dimensional rigid motions, combining a rotation and a translation with composition and inversion operations. It is the natural space for robot poses and frame transforms, but local coordinates and uncertainty on the group require careful conventions.
Also known as: 3D rigid-motion group
Updated
Treat poses as elements of a group
An element of SE(3) contains a three-dimensional rotation and translation. A 4 by 4 homogeneous transformation is a common matrix representation. Matrix multiplication composes frame changes, the identity represents no motion, and the inverse reverses a transform, so the set satisfies the operations of a Lie group.
Solà, Deray, and Atchuthan present the Lie-group tools used in robot state estimation and provide formulas for SE(3) and other common groups. The group structure preserves valid rotations when poses are composed, unlike treating all matrix entries as unrelated numbers.
Connect finite poses to local motion
The Lie algebra se(3) is the tangent space at the identity. Its six local coordinates describe an angular and linear twist. Exponential and logarithm maps connect a local perturbation with a finite pose, which supports interpolation, optimisation, and error definitions.
Pose estimation, visual odometry, SLAM, hand-eye calibration, and whole-body kinematics all combine uncertain transforms between coordinate frames. Working on SE(3) makes the composition order and geometry explicit. A rigid-body transformation is an element or action of this group, while SE(3) names the whole structured set.
Coordinates and conventions can still fail
SE(3) is not an ordinary six-dimensional vector space globally. Adding pose coordinates directly can leave the group, and different left- or right-sided perturbations produce different Jacobians and covariance meanings. The logarithm also has branch and conditioning issues near certain rotations.
Implementations must agree on active versus passive transforms, frame direction, multiplication order, quaternion convention, and whether perturbations act on the left or right. A valid SE(3) matrix says nothing about collision, reachability, timing, or physical uncertainty unless those are modelled separately.
Sources
Related terms
Rigid-body transformation
A rigid-body transformation changes a body's position and orientation without changing its shape or size. In three-dimensional robotics it consists of a proper rotation and a translation.
Homogeneous transformation
In rigid-body robotics, a homogeneous transformation is a 4-by-4 matrix that combines a three-dimensional rotation and translation. It represents a pose or changes coordinates between reference frames.
Twist
A twist is a six-component representation of a rigid body's instantaneous motion, combining angular and linear velocity. Its numerical values depend on the reference frame and the point used for the linear component.
Screw theory
Screw theory is a geometric framework for describing rigid-body motion and forces using axes, rotation, translation, and pitch. In robotics it provides the basis for twist and wrench representations and screw-axis formulations of kinematics.
Pose estimation
Pose estimation determines the position and orientation of an object or robot relative to a reference frame. For a rigid body in three-dimensional space, a full pose has three translational and three rotational degrees of freedom.