Lie Group

A mathematical object that is both a group and a smooth manifold; rotations and poses in robotics are both Lie groups.

A Lie group, named after the Norwegian mathematician Sophus Lie (1842–1899), is a group that is also a smooth manifold, where both multiplication and inversion are smooth operations. Being a group guarantees elements can be composed (two rotations combine into one) and inverted; being a manifold (a surface that looks locally like flat space) guarantees you can take derivatives and run optimization on it. Robotics relies on two above all: SO(3), the set of all 3D rotation matrices, and SE(3), rigid-body poses combining rotation and translation. Their tangent spaces at the identity element are called Lie algebras — so(3), made up of skew-symmetric matrices, corresponds to SO(3) — and the two convert into each other through the exponential map. This makes it possible to add, subtract, and take gradients using ordinary 3D vectors, then map the result back to a rotation while guaranteeing the result is still a valid one. SLAM, visual odometry, state estimation, pose optimization, and the product of exponentials formula are all built on this foundation.

ExampleWhen optimizing a camera's orientation in visual SLAM, the rotation matrix R's 9 numbers aren't updated directly; instead, a small 3D increment δ is solved for and applied as R ← R·exp([δ]×) ([δ]× is the skew-symmetric matrix built from δ, and exp is the matrix exponential), so the updated R is guaranteed to remain a valid rotation matrix.

Also called

Matrix Lie Group, Continuous Transformation Group

Related

Special Orthogonal Group SO(3) · Exponential Map · Skew-Symmetric Matrix · Adjoint Representation · Product of Exponentials Formula · Screw Theory

Sources

Wikipedia: Lie group
Solà et al., A micro Lie theory for state estimation in robotics (arXiv:1812.01537)
Lynch & Park, Modern Robotics（2017）第 3 章：SO(3)、SE(3) 与李代数 (Chinese)